Question

Gallup recently conducted a survey about the proportion of Americans who use e-cigarettes (vape). The following example is based on hypothetical data that is meant to be similar to the data collected by Gallup.

The following table considers survey data about annual household income and whether or not a person vapes.

	<$35,000	$35,000-$99,999	$100,000+	Total
Vape	47	57	19	123
Do Not Vape	381	659	362	1402
Total	428	716	381	1525

Problem 1. Based on recent data, about 28% of Americans earn less than $35,000 annually, about 42% of Americans earn between $35,000 and $99,999 annually, and about 30% of Americans earn more than $100,000. Does it appear that the sample is representative of the population? In other words, does it appear that the total people in each income category matches the appropriate proportion? Conduct a chi-square goodness of fit test at the 5% significance level by completing the following steps:

1. State the null and alternative hypotheses in words.
2. Compute the expected frequencies for each of the three category. Be sure to show your work.
3. Compute the test statistic using the observed frequencies from the table and the expected frequencies you computed in part (b).
4. State the degrees of freedom and find the critical value.
5. Answer the question: does it appear that the sample is representative of the population? Justify using either the critical value method or p-value method.

Answer 1

a.

Null hypothesis : H_o: The proportions of each category matches with the given population proportions

Alternate Hypothesis : H_a : The proportions of each category does not matches with the given population proportions

b.

Total number of Americans in the survey = 1525

Expected frequency for category of Americans earn less than $35,000 annually = 28% of 1525 =427

Expected frequency for category of Americans earn between $35,000 and $99,999 annually = 42% of 1525 = 640.5

Expected frequency for category of Americans earn more than $100,000 = 30% of 1525 = 457.5

c.

d. Degrees of freedom = Number of categories - 1 =3-1=2

Critical value of at 5%(:0.05) significance level for 2 degrees of freedom = 5.991

e.

As value of the test statistic : 21.6938 > Critical value of : 5.991. Reject the null hypothesis.

There is sufficient evidence to suggest that it does not appear that the samples is representative of the population.