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

[tex]
\[
\begin{tabular}{|l|r|r|r|r|}
\hline
\text{Preferred color} & \text{Black} & \text{White} & \text{Silver} & \text{Gold} \\
\hline
\text{Observed counts} & 10 & 10 & 4 & 9 \\
\hline
\end{tabular}
\]
[/tex]

The company wants to use these results to carry out a [tex]\chi^2[/tex] 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 :

To determine which counts make this sample fail the large counts condition for the [tex]\(\chi^2\)[/tex] goodness-of-fit test, we need to compare the expected counts with the observed counts for each color preference. The large counts condition typically requires that each expected count be at least 5, so we'll calculate the expected counts based on the given proportions and the total sample size.

1. Calculate Expected Counts:
- Total sample size: 33 customers

- Expected probability for each color:
- Black: 55%
- White: 20%
- Silver: 10%
- Gold: 15%

- Calculate expected counts based on these percentages:
- Expected count for Black: [tex]\(33 \times 0.55 = 18.15\)[/tex]
- Expected count for White: [tex]\(33 \times 0.20 = 6.6\)[/tex]
- Expected count for Silver: [tex]\(33 \times 0.10 = 3.3\)[/tex]
- Expected count for Gold: [tex]\(33 \times 0.15 = 4.95\)[/tex]

2. Check Large Counts Condition:
- Each expected count should be at least 5 for the test to be valid.

- Expected counts compared to the condition of 5:
- Black: 18.15 (okay)
- White: 6.6 (okay)
- Silver: 3.3 (fails the condition)
- Gold: 4.95 (fails the condition)

3. Identify which counts fail the large counts condition:
- The expected count for Silver is 3.3, which is less than 5. Therefore, Silver fails the large counts condition.
- The expected count for Gold is 4.95, which is also less than 5. Therefore, Gold fails the large counts condition.

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

- (B) The observed count of people who prefer silver.
- (D) The expected count of people who prefer silver.
- (1) The expected count of people who prefer gold.

These are the options that indicate failure of the large counts condition.

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