Question

Nationwide Insurance would like to perform a chi-square test to investigate whether a difference exists in the proportion of male and female teenagers who text while they drive. A random sample of 80 male teenagers found that 50 indicated they texted while driving. A random sample of 120 female teenagers found that 65 indicated they texted while driving. Using α = 0.5, the conclusion for this chi-square test would be that because the test statistic is

Answer 1

The chi-square test compares observed frequencies of male and female teenagers who text while driving with expected frequencies to determine if there is a difference in proportions. Calculating the chi-square statistic and comparing it with the critical value or p-value allows us to either reject or fail to reject the null hypothesis at the 5% significance level.

Step-by-step explanation:

The question is asking for the conclusion of a chi-square test to determine whether there is a significant difference in the proportion of male and female teenagers who text while driving, using a significance level (alpha) of 0.05. Based on the data provided, out of 80 male teenagers, 50 reported texting while driving, and out of 120 female teenagers, 65 reported texting while driving. We will use this information to calculate the expected frequencies under the null hypothesis that there is no difference between the proportions.

Chi-Square Test Calculation

First, we need to calculate the total number of teenagers who text while driving and the total number of participants. The sum of the males and females who text while driving is 50 + 65 = 115. The total number of participants is 80 + 120 = 200.

The expected frequency of male teenagers who text while driving if there were no difference would be the total who text (115) multiplied by the proportion of male teenagers in the survey (80/200), giving us an expected frequency of 46. The expected frequency of female teenagers who text while driving would be 115 multiplied by the proportion of female teenagers (120/200), giving us an expected frequency of 69.

With this data, we can compute the chi-square test statistic, compare it with the critical value from the chi-square distribution table for the appropriate degrees of freedom, and determine if we should reject the null hypothesis or not. If the p-value is less than alpha, we reject the null hypothesis.

If the conclusion is to reject the null hypothesis, then there is sufficient evidence to suggest that there is a difference in the proportion of males and females who text while driving. If we fail to reject the null hypothesis, we do not have evidence to suggest a difference.

Answer 2

Since the test statistic is less than the critical value while using α = 0.5, the conclusion for this chi-square test would be that we cannot reject the null hypothesis and cannot conclude that there is a difference in the proportion of male and female teenagers who text while they drive.

Each statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is correct is called a chi-squared test.

Through sample variance or from a sum of squared, chi-squared test are often constructed from them. Rising from an assumption of independent normally distributed data are test statistics that follow a chi-squared distribution, which is valid in a lot of cases due to the central limit theorem.