Question

QUESTION:

Sunlight and serotonin

Sunlight and darkness trigger the release of hormones in your brain. Exposure to sunlight is thought to increase the brain’s release of a hormone called serotonin. Serotonin is associated with boosting mood and helping a person feel calm and focused. At night, darker lighting triggers the brain to make another hormone called melatonin. This hormone is responsible for helping you sleep.

Without enough sun exposure, your serotonin levels can dip. Low levels of serotonin are associated with a higher risk of major depression with seasonal pattern (formerly known as seasonal affective disorder or SAD). This is a form of depression triggered by the changing seasons.

A mood boost isn’t the only reason to get increased amounts of sunlight. There are several other health benefits associated with catching moderate amounts of rays.

A study conducted at Dede University showed that hospital patients in rooms with lots of sunlight requires less medication for depression and during their total stay in the hospital than

patients who are in darker rooms. HADS-Depression measure is used to investigate hospital anxiety and depression, and 28 patients are randomly selected. 13 patients in the sunny rooms averaged 14 and gave a variance of 15 for their total stay as opposed to 15 patients in darker rooms averaged 12 and resulted an estimate of the variance of the population 25.3.

Assume that HADS-Depression level is normally distributed.

1. What is the variable of interest?                                                                                                                 
2. What is the point estimate for the mean HADS-Depression level of patients who stay in sunny rooms?          
3. A researcher believes that the mean of HADS-Depression level is more than 15.3 for the patients who stay in sunny rooms. Do you think that her claim is correct? Test the claim, at 5% level of significance.
   1. State the hypotheses.                                                                                                                       
   2. Write the test statistic and find its value.                                                                                         
   3. Define the rejection criteria by using critical value approach.                                                        
   4. Write a detailed interpretation of the test result.                                                                        
4. Construct a 99% confidence interval on the difference in mean HADS-Depression level of patients who stayed in sunny and darker rooms. Assume equal population variances. State your comment.                                               
5. Test at the 1% level of significance to see whether there is a significant difference (positive or negative) in mean HADS-Depression level of patients who stayed in sunny and darker rooms. Assume the equal population variances. Show all steps of the procedure.
   1. State the hypotheses.                                                                                                                       
   2. Compute necessary quantities.                                                                                                        
   3. Define the rejection criteria.                                                                                                            
   4. Write a detailed interpretation of the test result.
6. Find a 99% lower confidence bound on the difference in two population means. Interpret your result. Assume that population variances are different.                                                                                                                                 
7. Is there enough evidence to support the claim that the mean HADS-Depression level of patients who stayed in darker rooms is less than the mean HADS-Depression level of patients who stayed in sunny rooms? Assume that population variances are different. Use a=0.01.
   1. State the hypotheses.                                                                                                                        
   2. Write the test statistic and find its value.                                                                                          
   3. Define the rejection criteria by using critical value approach.                                                         
   4. Write a detailed interpretation of the test result.

Answer 1

We calculate the test statistic using the formula mentioned. The critical value is obtained from STATKEY (images attached for reference). We construct the C.I interval as shown. If the population variances are equal we use the pooled t test, otherwise we use the unpooled t test.