Math Core

This test covers chi-square goodness-of-fit tests and chi-square tests for homogeneity and independence: expected counts, the $\chi^2$ statistic, degrees of freedom, P-values, conditions and conclusions.

Problems 1–4 use this setting. A city's public works department claims that its repair calls are $50\%$ potholes, $30\%$ traffic signals and $20\%$ street signs. A reporter takes a random sample of $150$ recent repair calls and finds $93$ pothole calls, $33$ traffic-signal calls and $24$ street-sign calls.

Question 1

If the department's claim is true, what is the expected number of traffic-signal calls in the sample?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

Calculate the chi-square goodness-of-fit statistic for the reporter's sample.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 3

Find the P-value for the test using technology. Round to three decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

Which conclusion is correct?

Question 5

A manager plans a goodness-of-fit test using a random sample of $60$ of the store's many customers to test the claim that customers pay $55\%$ by card, $40\%$ by phone and $5\%$ by cash. Which statement about the conditions is correct?

Question 6

A researcher takes separate random samples of $200$ residents from each of four cities and asks each person which of five ways they most often get local news. She wants to know whether the distribution of news source differs among the cities. Which test should she use?

Question 7

A chi-square goodness-of-fit test with $4$ categories gives $\chi^2 = 10.1$ . Using a table of critical values, which range contains the P-value?

Problems 8–12 use this table. A random sample of $300$ students at a large university reported their daily caffeine use and their typical nightly sleep.

	More than 8 h	6–8 h	Less than 6 h	Total
No caffeine	$28$	$34$	$18$	$80$
Moderate	$40$	$52$	$48$	$140$
High	$22$	$24$	$34$	$80$
Total	$90$	$110$	$100$	$300$

Question 8

If caffeine use and sleep are independent, what is the expected count of students with moderate caffeine use who sleep 6–8 hours? Round to two decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 9

How many degrees of freedom does the chi-square test for this table have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

Calculate the chi-square test statistic. Round to two decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

The P-value for the caffeine test is about $0.111$ . Which conclusion is correct at $\alpha = 0.05$ ?

Question 12

Which cell contributes the most to the chi-square statistic, and how does its observed count compare with its expected count?

Question 13

In a randomized experiment, $62$ of $100$ plants given a new fertilizer flowered early, compared with $48$ of $100$ plants given the standard fertilizer. A two-sided two-proportion $z$ test gives $z \approx 1.99$ . What is the value of the chi-square statistic for a test of homogeneity on the same $2 \times 2$ table? Round to two decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

A chi-square test for homogeneity comparing favorite lunch options among three grades gives a P-value of $0.03$ . Which is the correct interpretation of this P-value?