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Math  /  Data & Statistics

QuestionA smart phone manufacturer is interested in constructing a 90\% confidence interval for the proportion of smart phones that break before the warranty expires. 92 of the 1683 randomly selected smart phones broke before the warranty expired. Round answers to 4 decimal places where possible. a. With 90%90 \% confidence the proportion of all smart phones that break before the warranty expires is between □\square and □\square b. If many groups of 1683 randomly selected smart phones are selected, then a different confidence interval would be produced for each group. About □\square percent of these confidence intervals will contain the true population proportion of all smart phones that break before the warranty expires and about □\square percent will not contain the true population proportion.

Studdy Solution

STEP 1

1. The sample size is n=1683 n = 1683 .
2. The number of smart phones that broke is x=92 x = 92 .
3. We are constructing a 90% 90\% confidence interval for the proportion.
4. The sample proportion p^ \hat{p} is calculated as xn \frac{x}{n} .
5. The confidence interval is calculated using the standard normal distribution (Z-distribution) for proportions.

STEP 2

1. Calculate the sample proportion.
2. Determine the critical value for a 90% confidence interval.
3. Calculate the standard error of the proportion.
4. Construct the confidence interval.
5. Interpret the confidence interval.

STEP 3

Calculate the sample proportion p^ \hat{p} :
p^=xn=921683≈0.0547 \hat{p} = \frac{x}{n} = \frac{92}{1683} \approx 0.0547

STEP 4

Determine the critical value z∗ z^* for a 90% confidence interval. For a 90% confidence level, the critical value z∗ z^* is approximately 1.645.

STEP 5

Calculate the standard error (SE) of the proportion:
SE=p^(1−p^)n=0.0547(1−0.0547)1683≈0.0055 SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.0547(1-0.0547)}{1683}} \approx 0.0055

STEP 6

Construct the confidence interval using the formula:
p^±z∗×SE \hat{p} \pm z^* \times SE
0.0547±1.645×0.0055 0.0547 \pm 1.645 \times 0.0055
Calculate the margin of error:
1.645×0.0055≈0.0090 1.645 \times 0.0055 \approx 0.0090
Calculate the confidence interval:
(0.0547−0.0090,0.0547+0.0090) (0.0547 - 0.0090, 0.0547 + 0.0090)
(0.0457,0.0637) (0.0457, 0.0637)

STEP 7

Interpret the confidence interval: a. With 90% 90\% confidence, the proportion of all smart phones that break before the warranty expires is between 0.0457 0.0457 and 0.0637 0.0637 .
b. If many groups of 1683 randomly selected smart phones are selected, then a different confidence interval would be produced for each group. About 90% 90\% of these confidence intervals will contain the true population proportion of all smart phones that break before the warranty expires, and about 10% 10\% will not contain the true population proportion.
The answers are: a. 0.0457 0.0457 and 0.0637 0.0637 b. 90% 90\% and 10% 10\%

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