1. Hand Calculation A medical researcher conjectured that heavy smoking is associated with prominent wrinkled skin around the eyes. The smoking habits

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1. Hand Calculation A medical researcher conjectured that heavy smoking is associated with prominent wrinkled skin around the eyes. The smoking habits as well as the prominence of wrinkles around the eyes were recorded on a random sample of 500 adults. Is there any evidence to support the conjecture, or are these two factors independent, using an α = 0.01 level of significance. Wrinkles Prominent Wrinkles not Prominent Heavy Smoker 95 55 Light or non-smoker 103 247

 (a) State the null and alternate hypothesis you would use.

(b) Calculate the expected value for each cell.

(c) Calculate the χ 2 test statistic for this test.

(d) State the degrees of freedom, and find the rejection region for this test.

(e) What can you conclude about the independence of the two factors? Give reasons.

 

2. Minitab Calculation The percentage of the population that is classified into different weight categories is compared for three countries. To do this three random samples of approximately the same size were sere selected from each population and the individuals classified into one of four weight categories. We want to determine if the proportions of those in each weight category is the same for all three countries, using α = 0.01 level of significance. Britain Canada USA Underweight 126 297 156 Normal 306 498 349 Overweight 88 61 75 Obese 27 17 44

(a) State the null and alternate hypothesis you would use.

(b) What are the values of the test statistic and the p-value.

(c) What can you conclude about the proportions of the population in each of the weight categories for the three countries? Say why in one sentence. 1 NOTE: In Minitab to do a Contingency Table, χ 2 test, for either independence, or equal proportions in different populations, first enter the table into Minitab spread sheet. Then use the following command: Stat -> Tables -> Chi-Squared Test(Two-Way Table in Worksheet)

 

3. Hand Calculation A random sample of 248 measurements were taken from a population thought to have a uniform distribution on the interval [−4, 4]. The measurements were divided into groups, the first group being values in the interval [−4, −3), the second group being values in the interval [−3, −2) etc. The number of measurements in each of these groups is indicated in the table below as the count. Group [-4, -3) [-3, -2) [-2, -1) [-1, 0) [0, 1) [1, 2) [2, 3) [3, 4] Counts 36 38 26 38 32 28 23 27 Perform a χ 2 goodness of fit test to see if there is any evidence that the data does not comes from a uniform distribution on [−4, 4], using α = 0.05.

(a) State the null and alternate hypothesis you would use.

(b) Calculate the expected values for this test.

(c) Calculate the χ 2 test statistic for this test.

(d) State the degrees of freedom, and find the rejection region for this test.

(e) What can you conclude about the distribution that is being sampled? Give a reason.


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