Sampling+Distribution+Problems

==**Sample Problems: ** ( To see answer, highlight black lines under the question) ==

= = =** Identify the type of sampling used: **= 1) Researchers waited outside a bar they had randomly selected from a list of such establishments. They stopped every 10th person who came out of the bar and asked whether he or she thought drinking and driving was a serious problem. Systematic Sampling (Convenience bias)

2) Hoping to learn what issues may resonate with voters in the coming election, the campaign director for a mayoral candidate randomly selects one block from each of the city's election districts. Staff members go there and interview all the residents they can find. Cluster Sampling (Convenience bias)

3) A business magazin mailed a questionnaire to the human resource director of all the Fortune 500 companies and received responses from 23% of them. (Nonresponse bias)

4) A question posted on a web site aske visitors to the site to vote for their favorite actor and actress. Convenience Sampling (Voluntary Response bias)

5) A large company wanted to survey its employees' level of job satisfaction. It randomly selected 5 employees for each department to interview. Stratified Sampling (Response bias)

=** Essay Problem 1: **=

A researcher wants to conduct a study to test whether listening to soothing music for 20 minutes helps to reduce diastolic blood pressure in patients with high blood pressure, compared to simply sitting quietly in a noise-free environment for 20 minutes. One hundred patients with high blood pressure at a large medical clinic are available to participate in this study.

(a) Propose a design for this study to compare these two treatments.

(b) The null hypothesis for this study is that there is no difference in the mean reduction of diastolic blood pressure for the two treatments and the alternative hypothesis is that the mean reduction in diastolic blood pressure is greater for the music treatment. If the null hypothesis is rejected, the clinic will offer this music therapy as a free service to their patients with high blood pressure. Describe Type I and Type II errors and the consequences of each in the context of this study, and discuss which one you think is more serious.

Part (a):

Approach 1: Paired Design Each subject will receive both treatments, with a suitable length of time between treatments. The order of the treatments will be randomly assigned to the subjects. For example, for each patient flip a coin to determine which treatment will be administered first. Measure diastolic blood pressure, then have the subject sit quietly for 20 minutes in either a noise-free environment or in a room where soothing music is played, depending on which treatment was selected at random (based on the coin flip). At the end of the 20 minutes, measure diastolic blood pressure again and compute its change ( // after − before) //. After a suitable period of time, repeat with the other treatment. When the data have been collected, the difference ( // music // − // noise-free // ) in the change in diastolic blood pressure will be computed for each subject, and then a paired // t // -test will be run to see if the mean difference is significantly different from zero.

Approach 2: Matched Pairs Design Measure diastolic blood pressure for each of the 100 subjects and then form 50 pairs based on these readings by pairing the two with the highest diastolic blood pressure, then the two with the next highest, and so on. For each pair, toss a coin to determine which member of the pair will be assigned to group 1, and then assign the other member of the pair to group 2. For group 1, measure diastolic blood pressure, then have the subjects sit quietly in a noise-free environment for 20 minutes, and then measure diastolic blood pressure again and compute its change ( // after − before) //. For group 2, the plan is the same, except that they will sit for 20 minutes in a room where soothing music is played between blood pressure measurements. When the data have been collected, the difference ( // music // − // noise-free // ) in the change in diastolic blood pressure will be computed for each pair, and then a paired // t // -test will be run to see if the mean difference is significantly different from zero.

Approach 3: Completely Randomized Design (This is not as good a choice as the two previous approaches, but because of the large number of subjects available for each treatment group, it is considered an acceptable solution.) Assign the 100 patients numbers from 00 to 99. From a random number table, select 50 unique numbers; the patients with the selected values will form group 1; the remaining 50 patients will form group 2. For group 1, measure diastolic blood pressure, then have the subjects sit quietly in a noise-free environment for 20 minutes, and then measure diastolic blood pressure again. For group 2, the plan is the same, except that they will sit for 20 minutes in a room where soothing music is played between blood pressure measurements. When the data have been collected, the change in diastolic blood pressure will be computed for each subject, and then a two-sample t-test will be run to see if there is a significant difference between the mean change attributable to // music // and the mean change attributable to a // noise-free // environment.

Part (b):

Type I error: Concluding that soothing music does reduce mean diastolic blood pressure compared to sitting quietly, when in fact it does not. The consequence of this type of error is that the clinic will offer music therapy when it is not effective.

Type II error: Soothing music does reduce diastolic blood pressure compared to sitting quietly, but we fail to detect this and conclude that it does not. The consequence of this type of error is that the clinic will choose not to offer music therapy when it would have been effective.

Which type of error is more serious? A case can be made for either type of error, and the student can take either side as long as a reasonable justification is given. For example, the student can say a Type I error is more serious because it will cost the clinic money with no benefit, or the student can say that a Type II error is more serious because the clinic will miss an opportunity to improve the health and well-being of its patients.

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