Statistics question

Question

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 36 arrests last month, 24 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

1. What is the value of the sample test statistic? (Round your answer to two decimal places.)
2. Find the P-value of the test statistic. (Round your answer to four decimal places.)

Answer 1

Question: Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 36 arrests last month, 24 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

1. What is the value of the sample test statistic? (Round your answer to two decimal places.)
2. Find the P-value of the test statistic. (Round your answer to four decimal places.)

Answer: Let's break this up a little bit. For a problem like this, identify statistics from research already carried out, and identify what you need to find in your current study.

Part 1: This is a hypothesis test in which we seek to draw a conclusion about p, the true proportion of males aged 15-34 years who were arrested in Rock Springs. The null is p = 0.7, and the alternative is p ≠ 0.7. This is the alternative because we are asked "to test the claim that the population proportion of such arrests in Rock Springs is different from 70%" (italics and bold print added for emphasis). The true proportion could be less than or greater than 0.7, so this is a two-tailed hypothesis test.

The formula for the test statistic is z₀ = (phat - p₀) / sqrt(p₀q₀/n). phat is our point estimate of p, which is 24 arrests of interest out of 36 files, or 24/36, or 2/3. This comes strictly from our research at hand, not the previous research. p₀ is the proportion that we are comparing against from the old research, or 0.7. In the bottom of the formula, q₀ comes from calculating 1 - p₀, which is 1 - 0.7 = 0.3. n is the number of files we're studying, or 36. Substituting our numbers into the formula we get:

z₀ = ((2/3) - (0.7)) / sqrt((0.7)(0.3)/36)

z₀ = (-1/30) / sqrt(.21/36)

z₀ = (-1/30) / sqrt(7/1,200)

z₀ ≈ -0.44

Part 2: Let's find the area associated with -0.44 as a z-quartile. Looking to a z-table, the area to the left of it (we'll check left since the z score is negative, and we have a rejection region split up in both tails of the distribution) is 0.3300. The p-value is the area to the left of z = -0.44 and to the right of z = 0.44. The distribution is symmetric, so multiply 0.3300 by 2 to arrive at the final p-value of 0.66. A calculator might be more precise with a value like 0.6625, and the values in a paper z-table are usually rounded to four decimal places. This problem also could have been done with the 1-PropZTest feature in a TI-83 or TI-84 calculator (use p₀ = 0.7, x = 24, n = 36, ≠).