Worked Example. So back to our example, if our previous example. If we determined that 7% of the 1000 sampled smoke, and we wanted to create 90% confidence interval, then we would perform the following steps: This means that we are 90% confident that the true proportion of smokers in the state is between …2.3 - Sample Size Needed for Estimating Proportion. Using the formula to find the sample size for estimating the mean we have: n = 1 d 2 z α / 2 2 ⋅ σ 2 + 1 N. Now, σ 2 = N N − 1 ⋅ p ⋅ ( 1 − p) substitutes in and we get: n = N ⋅ p ⋅ ( 1 − p) ( N − 1) d 2 z α / 2 2 + p ⋅ ( 1 − p) When the finite population correction ...A wonky affair that nonetheless serves to reflect the nation's changing tastes. Kids these days, with their melons and headphones! The UK’s statistics authority has announced the a... 1. To figure out part b, you need to find the 95th percentile. You can use a normal table to do this, find the z value which corresponds to 95% of the data to the left of the value. In your case, this z value is 1.645. Now you can use the formula x−μ σ = z = 1.645 x − μ σ = z = 1.645. 1. To figure out part b, you need to find the 95th percentile. You can use a normal table to do this, find the z value which corresponds to 95% of the data to the left of the value. In your case, this z value is 1.645. Now you can use the formula x−μ σ = z = 1.645 x − μ σ = z = 1.645. You want to know the proportion of slices remaining compared to the entire cake. You have 12 − 5 = 7 12-5 = 7 12 − 5 = 7 slices remaining, so the proportion is 7 / 12 7/12 7/12. If you wish to rewrite this proportion using a colon, you could write it as 7: 12 7:12 7: 12. Note that this proportion calculator will not …Steps. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability. On a normal distribution with a mean of 65 and standard deviation of 5, the proportion greater than 73 is 0.05480. In other words, 5.480% of vehicles will be going more than 73 mph. Learn how to calculate the sample size, the error bound, and the confidence interval for a population proportion using the normal distribution of proportions. See examples, formulas, and calculator solutions. A marginal distribution is simply the distribution of each of these individual variables. In a two-way table, the marginal distributions are shown in the margins of the table: For example, we would say that the marginal distribution of sports is: We could also write the marginal distribution of sports in percentage terms (i.e. out of the total ... Sensitivity is the proportion of people with the disease who will have a positive test result. The predictive value can be calculated from a 2×2 contingency table , like this one: The two pieces of information you need to calculate the positive predictive value are circled: the true positive rate (cell a) and the false positive rate (cell b). Learn how to calculate the sample proportion, the ratio of the number of successes in a sample to the size of the sample. See examples, formulas, and how to estimate the … Learn how to calculate the sample size, the error bound, and the confidence interval for a population proportion using the normal distribution of proportions. See examples, formulas, and calculator solutions. When you perform a hypothesis test of a single population proportion p, the steps are exactly the same as what we have seen before, however we will calculate our Test Statistic differently. When conducting a test for p, our hypotheses will look as follows: H o: p = p 0. H a: p (<,>,≠) p 0. Recall, the general form of a test statistic is:For this example question the X-value is your SAT score, 1100. Step 2: Put the mean, μ, into the z-score equation. Step 3: Write the standard deviation, σ into the z-score equation. Step 4: Find the answer using a calculator: (1100 – 1026) / 209 = .354. This means that your score was .354 std devs above the mean. 9.4 - Comparing Two Proportions. So far, all of our examples involved testing whether a single population proportion p equals some value p 0. Now, let's turn our attention for a bit towards testing whether one population proportion p 1 equals a second population proportion p 2. Additionally, most of our examples thus far have involved left ... Descriptive Statistics N Event Sample p 95% CI for p 1000 87 0.087000 (0.070268, 0.106208) In these results, the estimate of the population proportion for households that made a purchase is 0.087. You can be 95% confident that the population proportion is between approximately 0.07 and 0.106.A chi-square (Χ 2) goodness of fit test is a goodness of fit test for a categorical variable. Goodness of fit is a measure of how well a statistical model fits a set of observations. When goodness of fit is high, the values expected based on the model are close to the observed values. When goodness of fit is low, the values expected based on ...E B P = z α 2 × p ′ • q ′ n, where p′ is the sample proportion, q′ = 1 – p′, and n is the sample size. Solving for n gives you an equation for the sample size. n = ( z α 2) 2 ( p ′ q ′) E B P 2. This formula tells us that we can compute the sample size n required for a confidence level of Cl = 1 − α.Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-statistics/density-curves-n...where p p denotes the proportion of all adults who prefer the company’s beverage over that of its competitor’s beverage. Step 2. The test statistic (Equation 8.5.1 8.5.1) is. Z = p^ −p0 p0q0 n− −−−√ Z = p ^ − p 0 p 0 q 0 n. and has the standard normal distribution. Step 3. The value of the test statistic is.Need to justify that content marketing works? Discover the latest content marketing statistics. Find interesting facts, benchmark stats and metrics. Need to justify that content ma...What I want to introduce you to in this video is the notion of a proportional relationship. And a proportional relationship between two variables is just a relationship where the ratio between the two variables is always going to be the same thing. So let's look at an example of that. So let's just say that we want to think about the ...For this example question the X-value is your SAT score, 1100. Step 2: Put the mean, μ, into the z-score equation. Step 3: Write the standard deviation, σ into the z-score equation. Step 4: Find the answer using a calculator: (1100 – 1026) / 209 = .354. This means that your score was .354 std devs above the mean.Unit 8 Random variables and probability distributions. Unit 9 Sampling distributions. Unit 10 Inference for categorical data: Proportions. Unit 11 Inference for quantitative data: Means. Unit 12 Inference for categorical data: Chi-square. Unit 13 Inference for quantitative data: slopes.This release contains the latest statistics on school and pupil numbers and their characteristics, including age, gender, free school meals (FSM) …Sep 19, 2023 · Calculate basic summary statistics for a sample or population data set including minimum, maximum, range, sum, count, mean, median, mode, standard deviation and variance. Enter data separated by commas or spaces. You can also copy and paste lines of data from spreadsheets or text documents. See all allowable formats in the table below. Choosing the right chandelier size for your space is crucial to achieving a balanced and harmonious interior design. The wrong size can overpower a room or make it feel underwhelmi...Mar 13, 2018 · Add together the different sample sizes. January has 31 days and February has 28 days for the time period examined. Therefore, the total sample size is 59 days. Divide the number of times the event occurred by the total sample size to find the cumulative percentage. In the example, 25 days divided by 59 days equals 0.423729 or 42.3729 percent. The lower tail contains the lower values in a distribution. If you graph any distribution on a Cartesian plane, the lowest set of number will always appear on the left, because the lowest values on a number line are to the left. So, “lower tail” means the same thing as “left tail”.Sample statistical analysis is a crucial step in any research project. It involves examining a subset of data to make inferences about the larger population. However, there are sev...The formula for the confidence interval for a population proportion follows the same format as that for an estimate of a population mean. Remembering the sampling distribution for the proportion from Chapter 7, the standard deviation was found to be: σp' = p(1 − p) n− −−−−−−√ σ p' = p ( 1 − p) n. The confidence interval ...Statistical calculators, sample size, free, confidence interval, proportion, mean. Things to remember. A ratio is a comparison of two quantities. A proportion is an equality of two ratios. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed. A random sample of 25 statistics students was asked: “Have you smoked a cigarette in the past week?” Six students reported smoking within the past week. Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke. Solution Mar 13, 2018 · This is given by the formula Z= (X-m)/s where Z is the z-score, X is the value you are using, m is the population mean and s is the standard deviation of the population. Consult a unit normal table to find the proportion of the area under the normal curve falling to the side of your value. The left-hand column gives the z-score to a single ... We arbitrarily label one population as Population \(1\) and the other as Population \(2\), and subscript the proportion of each population that possesses the characteristic with the number \(1\) or \(2\) to tell them apart. We draw a random sample from Population \(1\) and label the sample statistic it yields with the … The Sampling Distribution of the Sample Proportion. For large samples, the sample proportion is approximately normally distributed, with mean μˆP = p and standard deviation σˆP = √pq / n. A sample is large if the interval [p−3 σˆP, p + 3 σˆP] lies wholly within the interval [0,1]. In this section, we begin by defining the point estimate and developing the confidence interval based on what we have learned so far. The point estimate for the difference between the two population proportions, \ (p_1-p_2\), is the difference between the two sample proportions written as \ (\hat {p}_1-\hat {p}_2\). The Formula for Percent Proportion is Parts /whole = percent/100. This formula can be used to find the percent of a given ratio and to find the missing value of a part or a whole. A percent proportion is an equation where a % of whole is equal to the part of the whole or we can say it is an equation where a percent is equal to an equivalent ratio. Jun 24, 2019 · Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke. Solution A Six students out of 25 reported smoking within the past week, so \(x = 6\) and \(n = 25\). Statistics How To has more than 1,000 articles and hundreds of videos for elementary statistics, probability, AP and advanced statistics topics. Looking for a specific topic? Type it into the search box at the top of the page. Latest articles. Markov’s Inequality; What is a Two Proportion Z-Interval? Bivariate Correlation and RegressionSo that would be our assumed population proportion times one minus our assumed population proportion divided by our sample size. And in future videos, we're gonna go all the away and calculate this, and then look it up in a z-table and see what's the probability of getting that extreme or more extreme of a result and compare it to …Oct 3, 2019 ... This statistics video tutorial provides the formulas for calculating the test statistic for the population mean and population proportion.Oct 26, 2020 · For these problems, it is important that the sample sizes be sufficiently large to produce meaningful results. The product of the sample size n and the probability p of the event in question occurring must be greater than or equal to 10, and similarly, the product of the sample size and one minus the probability of the event in occurring must also greater than or equal to 10. Apr 15, 2019 ... How about 100? 1000? You can take a guess, but statistics can turn that guess into (reasonably) solid numbers. We can't know the precise truth ...R software has become one of the most popular tools for statistical analysis in recent years. With its powerful features and user-friendly interface, it has become a favorite among...T-distribution and t-scores. A t-score is the number of standard deviations from the mean in a t-distribution.You can typically look up a t-score in a t-table, or by using an online t-score calculator.. In statistics, t-scores are primarily used to find two things: The upper and lower bounds of a confidence interval when the data are approximately …Proportion of people under a certain age: sample proportion: p-hat: ... Population parameters: Once the appropriate sample statistic is found, the population parameter can be estimated.The letter p is used for the population proportion, so you would write this fact like this: p = 237/1000. You can also write 237/100 as a decimal (by dividing 1000 by 237). If you did that, then p = 0.237. …A chi-square (Χ 2) goodness of fit test is a goodness of fit test for a categorical variable. Goodness of fit is a measure of how well a statistical model fits a set of observations. When goodness of fit is high, the values expected based on the model are close to the observed values. When goodness of fit is low, the values expected based on ...9.4 - Comparing Two Proportions. So far, all of our examples involved testing whether a single population proportion p equals some value p 0. Now, let's turn our attention for a bit towards testing whether one population proportion p 1 equals a second population proportion p 2. Additionally, most of our examples thus far … And then what would out standard deviation be for our sample proportion? Well, it's going to be equal to the square root of 0.6 times 0.4, all of that over 10. And we can get a calculator out to calculate that. So if we take 0.6 times 0.4 equals, divided by 10, equals, and then we take the square root of that, and we get it's approximately 0.15. The normal condition for sample proportions. According to the admissions director of a certain college, approximately 11 % of the 250 freshmen admitted had applied using the early decision option. Suppose that we took random samples of 20 freshmen from this population and computed the proportion p ^ of freshmen in each sample who had …To conduct the appropriate hypothesis test, choose the Stat > Proportion Stats > One Sample > With Summary menu option. In StatCrunch, a success is used to ...Apr 25, 2022 · A random sample of 29 statistics students was asked: “Have you smoked a cigarette in the past week?” Eight students reported smoking within the past week. Find a 95% confidence interval for the true proportion of statistics students who smoke. Solution. Eight students out of 29 reported smoking within the past week, so \(x = 8\) and \(n = 29\). Oct 26, 2020 · For these problems, it is important that the sample sizes be sufficiently large to produce meaningful results. The product of the sample size n and the probability p of the event in question occurring must be greater than or equal to 10, and similarly, the product of the sample size and one minus the probability of the event in occurring must also greater than or equal to 10. Let’s look at the 95% confidence intervals constructed via the formula by two surveyors: The first got a sample proportion of 0.38, and the second got a sample proportion of 0.34. Surveyor #1 Result. Using the formula with a sample proportion of 0.38 and a sample size of 1000, the 95 percent confidence interval is 0.35 to 0.41 (rounded).Jul 12, 2017 ... ... Statistics and Probability. I guess it's a standard thing which we just have to find somewhere on our own? ... For example, if you want to find ...Fisher’s Exact Test of Independence is a statistical test used when you have two nominal variables and want to find out if proportions for one nominal variable are different among values of the other nominal variable. For experiments with small numbers of participants (under around 1,000), Fisher’s is more accurate than the chi-square test ...Confirm that the sample is large enough to assume that the sample proportion is normally distributed. Use \(p=0.90\), corresponding to the assumption that the retailer’s claim is valid. Assuming the retailer’s claim is true, find the probability that a sample of size \(121\) would produce a sample proportion so low as was observed in … This way you know that p is the population proportion, and that \(\hat{p}\) is the sample proportion related to it. Now proportion tests are about looking for the percentage of individuals who have a particular attribute. You are really looking for the number of successes that happen. Thus, a proportion test involves a binomial distribution. Statistics How To has more than 1,000 articles and hundreds of videos for elementary statistics, probability, AP and advanced statistics topics. Looking for a specific topic? Type it into the search box at the top of the page. Latest articles. Markov’s Inequality; What is a Two Proportion Z-Interval? Bivariate Correlation and Regression While it is intuitively clear that the expected value of all sample proportions ought to be equal to the population proportion, it is helpful to understand why. First, for any constant c, E(cX) = cE(X). This follows because constants can be factored out of sums. The number 1=n is a constant, so. E(^p) = E. X 1 1. First, look at the left side column of the z-table to find the value corresponding to one decimal place of the z-score. In this case, it is 1.0. Then, we look up the remaining number across the table (on the top), which is 0.09 in our example. Using a z-score table to calculate the proportion (%) of the SND to the left of the z-score.Mar 13, 2018 · Add together the different sample sizes. January has 31 days and February has 28 days for the time period examined. Therefore, the total sample size is 59 days. Divide the number of times the event occurred by the total sample size to find the cumulative percentage. In the example, 25 days divided by 59 days equals 0.423729 or 42.3729 percent. To do so: Under the Stat menu, select Basic Statistics, and then select 2 Proportions...: In the pop-up window that appears, select Summarized data, and enter the Number of events, as well as the …Financial literacy in the U.S. leaves much to be desired, and our financial education statistics are bleak. Take a close look at the problem. While financial education statistics a...Nov 18, 2015 ... Find centralized, trusted content ... I was working on statistics using R. Before ... How do I find the proportion of a value in a variable on R?To determine whether the difference between the population proportion and the hypothesized proportion is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when ...Statistical summaries like proportions and means arising from random samples tend to hone in on the true population value. Further, as we saw in figure 8.1, a frequency curve (probability histogram) showing the distribution of a sample proportion or mean across all possible samples follows the normal curve. Here's the rule: Normal Approximation: 9.4 - Comparing Two Proportions. So far, all of our examples involved testing whether a single population proportion p equals some value p 0. Now, let's turn our attention for a bit towards testing whether one population proportion p 1 equals a second population proportion p 2. Additionally, most of our examples thus far have involved left ... Proportion is a fundamental concept in statistics that describes the percentage, fraction, or ratio of an element or a subset within a larger group. Learn four …For a proportion, the appropriate standard deviation is. √pq n. However, in the error bound formula, we use. √ˆpˆq n. as the standard deviation, …The confidence interval for a proportion follows the same pattern as the confidence interval for means, but place of the standard deviation you use the sample proportion times one minus the proportion: ... The test statistic is a number, calculated from a statistical test, used to find if your data could have … This sample proportion is written as p̂, pronounced p-hat. It’s calculated in the same way, except you use data from a sample: just divide the total number of items in the sample by the number of items you’re interested in. Example question: In a survey of 3121 people, 412 are under-vaccinated. Eigenvalue / number of variables = Proportion Variance. The higher the sum of square (SS) loadings (i.e., the eigenvalue), the more the factor is helping to explain the variances in the variables. According to the Kaiser Rule, a rule of thumb for interpreting the usefulness of factors, a factor is important if its eigenvalue is greater than 1. Looking for elementary statistics help? You’ve come to the right place. Statistics How To has more than 1,000 articles and hundreds of videos for elementary statistics, probability, AP and advanced statistics topics. Looking for a specific topic? Type it into the search box at the top of the page. Latest articles. Markov’s Inequality 2.3 - Sample Size Needed for Estimating Proportion. Using the formula to find the sample size for estimating the mean we have: n = 1 d 2 z α / 2 2 ⋅ σ 2 + 1 N. Now, σ 2 = N N − 1 ⋅ p ⋅ ( 1 − p) substitutes in and we get: n = N ⋅ p ⋅ ( 1 − p) ( N − 1) d 2 z α / 2 2 + p ⋅ ( 1 − p) When the finite population correction ... Looking for elementary statistics help? You’ve come to the right place. Statistics How To has more than 1,000 articles and hundreds of videos for elementary statistics, probability, AP and advanced statistics topics. Looking for a specific topic? Type it into the search box at the top of the page. Latest articles. Markov’s Inequality Add together the different sample sizes. January has 31 days and February has 28 days for the time period examined. Therefore, the total sample size is 59 days. Divide the number of times the event occurred by the total sample size to find the cumulative percentage. In the example, 25 days divided by 59 days equals 0.423729 or 42.3729 …Confirm that the sample is large enough to assume that the sample proportion is normally distributed. Use \(p=0.90\), corresponding to the assumption that the retailer’s claim is valid. Assuming the retailer’s claim is true, find the probability that a sample of size \(121\) would produce a sample proportion so low as was observed in … The sample proportion (p̂) describes the proportion of individuals in a sample with a certain characteristic or trait. To find the sample proportion, divide the number of people (or items) who have the characteristic of interest by the total number of people (or items) in the sample. For example, let’s say you were conducting a survey of 100 ... Oct 26, 2020 · For these problems, it is important that the sample sizes be sufficiently large to produce meaningful results. The product of the sample size n and the probability p of the event in question occurring must be greater than or equal to 10, and similarly, the product of the sample size and one minus the probability of the event in occurring must also greater than or equal to 10. ... find out. We only have p hat as an estimate of p ... And we already have the tools in statistics ... proportion is within two standard deviations of our sample ...9.4 - Comparing Two Proportions. So far, all of our examples involved testing whether a single population proportion p equals some value p 0. Now, let's turn our attention for a bit towards testing whether one population proportion p 1 equals a second population proportion p 2. Additionally, most of our examples thus far …Relative frequencies can be written as fractions, percents, or decimals. To find the relative frequency: For example, if three students in Mr. Ahab’s English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = = = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.Calculating a z interval for a proportion. Fatimah wants to estimate the proportion of students at her school who ate breakfast every day last week. She takes an. of 80 of the 1000 students at her school and finds that 20 of those sampled ate …Science requires that we make guesses, which is why we have confidence intervals. Advertisement Statistics is a bit of a mix between mathematics and probability. The point of stati...To conduct the appropriate hypothesis test, choose the Stat > Proportion Stats > One Sample > With Summary menu option. In StatCrunch, a success is used to ...How to find a population proportion is an essential skill in statistics. Jenn, Founder Calcworkshop ®, 15+ Years Experience (Licensed & Certified …Formula for calculating the probability of certain outcomes for an event. In this case: Probability of a coin landing on heads. Probability of an event = (# of ways it can happen) / (total number of outcomes) P (A) = (# of ways A can happen) / (Total number of outcomes) Example 1. There are six different outcomes.
find percentages of different quantities. calculate percentage increases and decreases. 1 Ratio. 1.1 Introduction. Ratios crop up often in official statistics.. Lucid mattress reviews
1. Checking the Conditions. The conditions for calculating a confidence interval for a proportion are: The sample is randomly selected. There is only two options: …The proportion difference of .08 (.52-.44) equates to 1.136 Standard Errors. Looking now at the Standard Normal Distribution (z-distribution) in Figure 10.2 we can see that the result is inside the 95% interval and so we do not reject the Null Hypothesis that Flowing Wells and Artesian Wells have equal population proportions. The Formula for Percent Proportion is Parts /whole = percent/100. This formula can be used to find the percent of a given ratio and to find the missing value of a part or a whole. A percent proportion is an equation where a % of whole is equal to the part of the whole or we can say it is an equation where a percent is equal to an equivalent ratio. Pooled estimate of a proportion ... Here x1 x 1 represents the number of successes in sample 1. x1 x 1 can be computed as ^p1n1 p ^ 1 n 1 if it is unknown.Jan 10, 2021 · The sample proportion for Connecticut is 239/2049 (11.7%) The sample proportion for New York = 9/550 (1.6%). The samples are different, but are they statistically different? a) Test with a significant level of 0.05 that the proportions of car with only rare license plate in Connecticut and New York are the same. In this Statistics 101 video, we learn about the fundamentals of sample proportions. To support the channel and signup for your FREE trial to The Great Cours... Sampling Distribution of P (Blue) The true proportion is p = P ( B l u e) = 2 5. When the sample size is n = 2, you can see from the PMF, it is not possible to get a sampling proportion that is equal to the true proportion. Although not presented in detail here, we could find the sampling distribution for a larger sample size, say n = 4.Answer: To find the proportion in statistics, divide the count of occurrences of a particular outcome by the total number of observations. In …Figure 19.4.1 19.4. 1: Distribution of leniency scores. One way to measure the effect of conditions is to determine the proportion of the variance among subjects' scores that is attributable to conditions. In this example, the variance of scores is 2.794 2.794. The question is how this variance compares with what the variance would have been if ...Unit 8 Random variables and probability distributions. Unit 9 Sampling distributions. Unit 10 Inference for categorical data: Proportions. Unit 11 Inference for quantitative data: Means. Unit 12 Inference for categorical data: Chi-square. Unit 13 Inference for quantitative data: slopes.Descriptive Statistics N Event Sample p 95% CI for p 1000 87 0.087000 (0.070268, 0.106208) In these results, the estimate of the population proportion for households that made a purchase is 0.087. You can be 95% confident that the population proportion is between approximately 0.07 and 0.106. When the study was complete, 588 teens had answered the question about their Facebook friends with 159 saying that they have more than 500 friends. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens that would report having more than 500 Facebook friends based on this larger sample. Need to justify that content marketing works? Discover the latest content marketing statistics. Find interesting facts, benchmark stats and metrics. Need to justify that content ma...Feb 24, 2023 · For example, write down this proportion, then draw one line between the purple terms, and another line between the green terms: 2. Multiply the two numbers connected by a line. One of the lines will connect two numbers (instead of a number and a variable like ). Find the product of these two numbers: 3. The One-Proportion Z-Test is a statistical test used to determine if the proportions of categories in a single qualitative variable significantly differ from an expected or known population proportion. To use it, you should have one group variable with only two options and you should have more than 10 values in every cell. See more below.Working from home or remotely has not lost steam. Here are important remote work statistics to know for 2023. Human Resources | Statistics WRITTEN BY: Charlette Beasley Published M...Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke. Solution A. Solution A Six students out of 25 reported smoking within the past week, so x = 6 and n = 25. Because we are using the “plus-four” method, we will use x = 6 + 2 = 8 and n = 25 + 4 = 29.Since COVID-19 began, the landscape of the moving industry has been changing. We will help clarify things with our article on must-know moving industry facts. Expert Advice On Impr... How to Find a Sample Size in Statistics. A sample is a percentage of the total population in statistics. You can use the data from a sample to make inferences about a population as a whole. For example, the standard deviation of a sample can be used to approximate the standard deviation of a population. Finding a sample size can be one of the ... .