A health researcher wanted to know if there was a difference in success rates between three methods of quitting smoking: A, B, and C. She designed a study with 15 participants, 5 in each group, and asked participants to track how many cigarettes they smoked in a 2-week period. In the results of her study, she reported as follows: “The means of the groups (Method A: M = 156, SD = 33.6; Method B: M = 163, SD = 34.4; Method C: M = 145, SD = 48.4) did not differ significantly, F(2, 12) = .27; ns.”

1)Provide an example of a study that would utilize a t test for a single sample. List three differences in how you would carry out hypothesis testing for a t test for a single sample versus for a Z test

An example of a study that would utilize a t test for a single sample is investigating whether the new weight loss program significantly reduces the average weight of participants after 12 weeks.

Three differences in carrying out hypothesis testing for a t test for a single sample versus a Z test are:

1) Sample size: For a t test, the sample size is typically smaller, whereas for a Z test, a larger sample size is required.
2) Use of the t-distribution: A t test uses the t-distribution, which accounts for the uncertainty in the population standard deviation. In contrast, a Z test assumes the population standard deviation is known or the sample size is large enough for the central limit theorem to be applied.
3) Assumptions: A t test assumes that the population of interest is normally distributed, whereas a Z test does not require this assumption if the sample size is large enough. Additionally, a t test assumes random selection and independence of observations, like a Z test.

An example of a study that would utilize a t-test for a single sample is to assess whether a new teaching method improves student performance in a particular subject.

Here are three differences in how you would carry out hypothesis testing for a t-test for a single sample compared to a Z-test:

1) Sample size: In a t-test for a single sample, the sample size is typically smaller compared to a Z-test. This is because the t-distribution is more sensitive to small sample sizes and accounts for the uncertainty introduced by estimating the population standard deviation from the sample.

2) Assumption about the population standard deviation: In a t-test for a single sample, the population standard deviation is unknown and needs to be estimated from the sample. Assumptions about the shape and variability of the population are not required, as the t-distribution is robust to departures from normality for large enough sample sizes.

3) Calculation of the test statistic: For a t-test, the test statistic is calculated using the sample mean, the estimated population standard deviation (sample standard deviation), and the sample size. The t-statistic is given by the formula: t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). This test statistic is then compared to the critical values from the t-distribution instead of the normal distribution used in a Z-test. Furthermore, the degrees of freedom used in the t-distribution calculation depend on the sample size (n-1 for a single sample t-test).

By understanding these differences, you will be able to appropriately apply and interpret the results of a t-test for a single sample in your study.