Probability and Statistics: Hypothesis Testing Two Samples and Paired Data

Company XYZ manufactures laptops. For quality control, two sets of laptops were tested. In the first group, 32 out of 800 were found to contain some sort of defect. In the second group, 30 out of 500 were found to have a defect. Is the difference between the two groups significant? Use a significance level of 0.05.

Patients were randomly assigned to either duct tape treatment or traditional treatment. Out of 100 warts treated with liquid nitrogen, 60 were successfully removed, and out of 104 warts treated with duct tape, 88 were successfully removed. Test if duct tape is more effective for removing warts at $\alpha = 0.01$.

Is there a difference between the variances of the number of weeks on the bestseller list for nonfiction and fiction books?

15 New York Times bestselling fiction books had a standard deviation of 6.17 weeks on the list. 16 New York Times best-selling nonfiction books had a standard deviation of 13.12 weeks.

At the 10% significance level, can we conclude there is a difference in the variances?

Conduct a hypothesis test to determine if a low-fat diet leads to greater weight loss compared to a control group, with a significance level of 5%.

Helen wishes to know whether giving away free stickers will increase her chocolate sales. She tests this by offering free stickers on some days and not on others, then compares the mean sales of the two groups using a t-test for the difference of two means.

There is no difference in birth weight between babies born in Birmingham hospitals ($\mu_B$) and babies born in Nottingham hospitals ($\mu_N$). However, the alternative hypothesis is that there is a difference. Define the null and alternative hypotheses.

A study was conducted to determine the effectiveness of a weight loss program. The table below shows the before and after weights of 10 subjects in the program. Is this program effective for reducing weight? Use a 5% significance level.

Calculate the t-value for a paired t-test given the before values, after values, and differences. The mean of the differences is 3, the standard deviation is 4.69, and there is a sample size of 8.

Using the MENT scores, test if there is a significant difference between veterans seeking compensation and those seeking treatment for post-traumatic stress disorder using pooled variance t-test and confidence intervals.

Test whether there is a significant difference in behavioral shame scores between violent and non-violent young offenders using a pooled-variance t-test.

Kaito takes a random sample of tomato plants from two separate fields and wants to conduct a two-sample t-test to see if the sizes of the plants differ between the fields. Assume all conditions for inference are met, with a significance level of 0.05.

Construct the null hypothesis that there is no difference between the mean sizes in the two fields (mean size in field A is equal to mean size in field B), and the alternative hypothesis that they differ (mean size in field A is not equal to mean size in field B).

One group was given vitamin B12 and one group was not, and then their times were measured when they ran a 50-yard dash. Perform a two-sample t-test to determine if there is a significant difference between the means of the two groups.