Lab - Single Sample t-Test

PSY 302 Lab Single-Sample t-Tests When to use a Single-Sample t-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) but an unknown standard deviation (  ) z -Test versus t -Test We use z -tests to test if the sample mean drawn from one population is different from that of a population with a known mean and standard deviation. In practice, however, we rarely actually have this information, and so we cannot conduct a z -test. The single sample t -test was developed to use when we do not know the population standard deviation and instead have to infer it based on the sample. Formula for a t-test Explanation t = M − μ s M t = Sample Mean - Population Mean Estimated Standard Error or t = Observed difference between sample and population estimated standard difference expected by chance Compared to a z-test, there are two differences: 1. We use sample standard deviation to calculate standard error. Notice that the t -test formula is just like the z -test formula with one difference: we use s M instead of  M . z -test t- test Population standard deviation is known. Population standard deviation is not known. σ M = σ √ N ; σ = √ ∑ ( X − μ ) 2 N s M = s √ N ; s = √ ∑ ( X − M ) 2 N − 1 z = M − μ σ M t = M − μ s M

2. Because sample s M is only an estimate, not the ground truth, we account for the uncertainty of that estimate by using t-distributions instead of a z-distribution. The main difference is that t-distributions have heavier tails than the normal distribution, especially for small sample sizes. Thus, you need a more extreme t-value than you would need a z-value, to reject the null. Conducing a Single Sample z-Test By Hand Example : In college many students tend to be away from their families for the first time. You hypothesize that this may make students feel lonelier than the general population. In order to answer the question of whether students are lonelier than the general population, you randomly select 20 students at the UO and ask them to report how lonely they are using a 1-10 Likert scale with 1 representing low loneliness and 10 representing high loneliness. You choose a commonly employed scale. Years of data collection with this scale in other populations have shown that mean loneliness in the general population is 3.70 ( µ = 3.70) . The standard deviation for loneliness in the general population is unknown . Null hypothesis: The average loneliness of UO students is equal to 3.70. Alternative hypothesis: The average loneliness of UO students is not equal to 3.70. Loneliness ratings on 1-10 scale of 20 college students at the UO: 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 6, 7, 7, 7, 8, 8, 9 Step 1 : Pick the test statistic and check assumptions.  Which test should we use?  What are the tests’ assumptions?

Step 4. Calculate the test statistic. Follow the steps outlined below: 1. Calculate the sample mean and standard deviation. X X - M (X - M) 2 1 1

1 1 2 2 2 2 3 3 3 3 4 6 7 7 7 8 8 9 M = SS = s = √ SS n − 1 = 2. Calculate the standard error of the mean. s M = s √ N = 3. Calculate the test statistic for the single-sample t-test. t = M − μ s M = ¿ Step 5. Decide whether to reject or fail to reject the null hypothesis.  Using absolute values (i.e., ignoring ‘-’ signs), is the critical value bigger or smaller than the test statistic?  What is your decision about the null hypothesis?

Move the Loneliness variable to the Dependent Variable section.  In the hypothesis section, o Make sure you enter the known population mean for the test value (this is how you specify that you want to compare your sample mean to the known population mean) o For a two-tailed test, select ≠ Test value (this is the default)  Select the following additional statistics: o Mean difference o Confidence interval o Effect size o Descriptives o Descriptives plots Let’s interpret each part of the output in Jamovi. Then, we will summarize all of the output in a concise APA-style summary. State your decision about the null hypothesis and your rationale for this decision: Write an APA-style summary of the results: