Lab - Single Sample z-Test

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Feb 20, 2024

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PSY 302 Lab Single Sample z-Test When to use a Single-Sample z-Test? - When you have a research question that can be answered by comparing a single sample to a population, and the population has a known mean (µ) and standard deviation (  ) - A single-sample z-test is infrequently used because we don’t often have population- level data with a known µ and  Jamovi actually doesn’t have a single-sample z-test feature. There are online calculators you can go to in order to perform a single-sample z-test, like this one: https://www.socscistatistics.com/tests/ztest_sample_mean/default2.aspx Today, we will compute one z-test by hand and compare our results to the online calculator Conducting a Single Sample z-Test By Hand Example: A researcher is interested in whether the extraversion levels of Canadians differs from the extraversion levels of people living in the United States. In order to test this hypothesis, the researcher collects extraversion data from a sample of 15 Canadians. The participants’ scores were: 1, 5, 4, 3, 4, 4, 5, 2, 2, 1, 5, 3, 3, 4, 5 The average extraversion score for the sample of Canadians is 3.40 ( M = 3.40, SD = 1.40). The researcher knows that the average extraversion score on this scale for the entire population of Americans is 3.70 (  = 3.70 ) and the known population standard deviation is 1.10 (  = 1.10 ). Determine whether the extraversion levels of Canadians differs significantly from the average extraversion of Americans using an alpha of .05 and a two-tailed test . Step 1: Pick the right test and check the assumptions. We are comparing a single sample to the population with known parameters (µ and  ). This is a job for the single-sample z-test.

Each type of statistical test makes certain assumptions. These are the assumptions for the single sample z-test:  Random sample and normality are robust assumptions, meaning that the test is still valid even if these assumptions are somewhat violated. For example, if the population distribution was skewed rather than normal, you can still do the test, as long as your sample size is large enough. That’s because of central limit theorem – even if the population distribution is not normal, the sampling distribution will be normally distributed when the sample size is sufficiently large (n ≥ 30).  Independence of observations is not robust . For example, if some participants end up chatting with each other while filling out your questionnaire and decide to submit matching answers, their data are no longer independent and you cannot perform the analysis. Step 2: State the Null and Alternative Hypotheses The null hypothesis (H 0 ) states the expected result if an independent variable has no effect on a dependent variable: H 0 :  Canadians = 3.70 Null stated in words: the average extraversion of Canadians is equal to 3.70 The alternative hypothesis (H 1 ) states the expected result if the independent variable does have an effect on the dependent variable: H 1 :  Canadians  3.70 Alternative stated in words: the average extraversion Note : Notice that the hypotheses are statements about a population .

Step 3: Setting a Decision Rule a) Specify alpha (alpha is your willingness to make a Type 1 error) (false positive) α = 0.05  Alpha: the researcher’s willingness to make a Type I error (typically set to 5% in psychological research)  Type 1 Error: Concluding that there is an effect when, in reality , there is no effect b) Construct a sampling distribution of means that represents the results you would expect to obtain if the null hypothesis were true . Draw the sampling distribution here. The value that you set alpha to determines where the rare zone and common zone are on your sampling distribution of means. You can find where these zones are on your distribution by finding the critical values . c) Find the critical values corresponding to your cut-off points. We most often conduct so- called two-tailed tests , meaning that the most extreme 5% of possible sample values (the rare zone) is evenly distributed between the left tail (extremely low values) and the right tail (extremely high values). Thus, when you look up the critical z-value in the z-table, you would find the z-value that corresponds to 97.5 percentile (rather than 95 th percentile). You already know that 95% of values in the normal distribution is confined between z=-1.96 and z=+1.96. If you needed the critical value for a different alpha level (e.g., alpha = 0.01), you would find the z-value that differentiates the middle 99% of values form the 0.5% of highest and 0.5% of

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lowest values. Label the upper and lower critical values on the sampling distribution you drew above. Step 4: Calculate the test statistic for the single sample z-Test The formula for calculating the single sample z-Test’s test statistic is: First, let’s calculate the z-statistic “by hand”, given the information provided in the example. Sample mean, population mean and population standard deviation have been already given to you. If the sample mean wasn’t calculated for you, you can always get the sample mean using Jamovi (go to Exploration  Descriptives and check off Mean). To calculate the z-statistic associated with the sample mean, you first need to know how much the sample means are expected to vary around the true mean. In other words, you need to calculate the standard error, or the typical sampling error you expect. σ M = σ √ n σ M = 1.10 √ 15 = 0.2840 From there, you can calculate the z-statistic. How far is the sample mean from the hypothesized mean, in units of standard error? z = 3.40 − 3.70 1.40 =

Step 5: Decide whether to Reject the Null Hypothesis or Fail to Reject the Null Hypothesis You can now compare your observed z-statistic to the sampling distribution and the critical values you draw earlier. Does it fall into the common zone or the rare zones? The z-test statistic we obtained falls in the common zone How do you decide if your test is statistically significant and you should reject the null? - If the test statistic lands in the rare zone , OR if the p-value is less than .05: o Decision: Reject the null hypothesis - If the test statistic lands in the common zone , OR if the p-value is greater than .05: o Decision: Fail to reject the null hypothesis NOTE : Either method (comparison to critical values or obtaining an exact p-value) should give you the same answer Our decision is fail to reject the null hypothesis because p>0.05. There is a greater than 5% chance of obtaining a z-test statistic when the null hypothesis is true.

Conducting a Single Sample z-Test Using a Statistical Software Program Now let’s navigate to the online calculator that we can use to calculate the test statistic for the single sample z-test using a computer: https://www.socscistatistics.com/tests/ztest_sample_mean/default2.aspx 1. To perform the single sample z-test using this program, we just need to enter the population mean (µ), population variance ( σ 2 ), sample mean ( M ), and sample size ( n ). 2. Then, select the correct alpha level (called significance level by this program) and specify whether you are conducting a one-tailed or two-tailed test. We typically set alpha to 0.05 and perform two-tailed tests. 3. Click on Calculate z-Score. Did you get the same answer as you did when conducting the single sample z-test by hand? Now let’s write up our conclusions based on these results. Non-APA statement of decision about the null State your decision about the null and your rationale for making this decision: Null hypothesis: The average extraversion of Canadians is equal to 3.70. Decision and Rationale: Fail to reject the null hypothesis because p>0.05 Write an APA-style summary of the results See the “APA-Style Formatting Guidelines” document on Canvas for guidelines on how to write up an APA-style report. Using a single sample z-test, we found that the average extraversion of Canadians (M=3.40, SD= 1.40), was lower, but not significantly different from, the average extraversion of Americans (  =3.70,  =1.10), z(n=15), =1.06, p=0.289.

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