Question

5) The police think that the true murderer is extremely smart. Examine the information below concerning test scores. You can safely eliminate any suspect who did not score in the 98^th percentile. For each suspect, sketch a graph of the distribution and the approximate location of the score to help you compute the percentile.

(a) The crazy mom scored a 168 on the quantitative portion of the GRE. GRE scores are normally distributed with a mean quantitative score of 151.3 and a standard deviation of 7.7.

(b) The neighbor with a stockpile of toilet paper scored a 770 on the GMAT. GMAT scores are normally distributed with a mean score of 563 and a standard deviation of 94.

(c) The self-proclaimed social media influencer scored 125 on a free online IQ test that was advertised on Facebook. IQ test scores are normally distributed with a mean score of 100 and a standard deviation of 15.

d) The police believe that the murder, which took place Tuesday afternoon, must have occurred in a location with very few visitors since no one has come forward as a witness. You can eliminate a location if the mean number of visitors on a Tuesday afternoon is greater than 15. Perform a hypothesis test for each possible murder location with α = 0.05. For simplicity, assume that all assumptions are satisfied.

The statistics collected from 10 Monday afternoons at each of the possible murder locations are recorded below.

(e) The abandoned X-bar Pool Hall: x̄ = 17, s = 3.8

(f) The Two-Tailed Tower: x̄ = 16, s = 1.1

(g) The Chi-Square town square: x̄ = 16, s = 2.2

Answer 1

We need to compute the p-value for all parts based on observed statistic. For first three problems the benchmark is 98 percentile, i.e. p-value less than 0.02 would be the rejection region

(a) The z score for crazy mom's score is

p-value is less than 0.2, so she is a suspect

(b) The z score for neighbor's score is

p-value is less than 0.2, so neighbor is a suspect

(c) The z score for social media influencer's score is

p-value is more than 0.2, so social media influencer is not a suspect

(d) For the next three parts, We now need to compute the p-value for population mean greater than 15, based on observed value, and the significance level is 0.05

(e) The t-statistic needs to be computed for n-1 = 9 degrees of freedom

p-value is more than 0.5, so abandoned X-bar Pool Hall is not a possible murder location

(f) The t-statistic needs to be computed for n-1 = 9 degrees of freedom

p-value is less than 0.5, so Two-Tailed Tower is a possible murder location

(g) The test-statistic is

p-value is more than 0.5, so Chi-Square town square is not a possible murder location