Question

A company plans on offering a new smartphone in four colors: black, white, silver, and gold. They suspect that 55% of customers prefer black, 20% prefer white, 10% prefer silver, and 15% prefer gold. They take a random sample of 33 potential customers to see what color they prefer. Here are the results:

Preferred color Black White Silver Gold
Observed counts 10 10 4 9 The company wants to use these results to carry out a x^2 goodness-of-fit test to determine if the sample disagrees with the expected distribution. Which count(s) make this sample fail the large counts condition for this test? Choose 2 answers: a. The observed count of people who prefer white. b. The observed count of people who prefer silver. c. The observed count of people who prefer gold. d. The expected count of people who prefer silver.
e. The expected count of people who prefer gold.

Answer 1

The χ2 goodness-of-fit test requires expected counts in each category to be at least five. In this case, the expected counts for silver (3.3) and gold (4.95) both fail to meet this condition, hence indicating that the sample fails the large counts condition for those colors.

Step-by-step explanation:

The company is planning to offer a new smartphone in four colors and wants to carry out a χ2 goodness-of-fit test to determine if the sample distribution of customer color preferences agrees with their expected distribution.

In a χ2 goodness-of-fit test, one condition for validity is that the expected counts in each category need to typically be at least 5 to ensure the accuracy of the test. The expected counts are calculated by multiplying the total sample size by the expected proportion for each category. Therefore:

• For black, the expected count is 33 * 0.55 = 18.15
• For white, the expected count is 33 * 0.20 = 6.6
• For silver, the expected count is 33 * 0.10 = 3.3
• For gold, the expected count is 33 * 0.15 = 4.95

Therefore, the counts that make this sample fail the large counts condition are:

• d. the expected count of people who prefer silver
• e. the expected count of people who prefer gold