Math / Data & Statistics

QuestionState whether each of the following changes would make a confidence interval wider or narrower. (Assume that nothing else changes.) a. Changing from a $95 \%$ confidence level to a $90 \%$ confidence level. b. Changing from a sample size of 400 to a sample size of 30 . c. Changing from a standard deviation of 30 pounds to a standard deviation of 15 pounds.
Click the icon to view the $t$ -table. a. How will changing from a $95 \%$ confidence level to a $90 \%$ confidence level affect the width of the confidence interval? A. The interval will become wider. B. The interval will become narrower. C. This change will not affect the width of the interval.

Studdy Solution

STEP 1

1. The confidence interval is calculated using the formula: $\bar{x} \pm t \cdot \frac{s}{\sqrt{n}}$ , where $\bar{x}$ is the sample mean, $t$ is the t-value, $s$ is the standard deviation, and $n$ is the sample size.
2. The width of the confidence interval is influenced by the confidence level, sample size, and standard deviation.

STEP 2

1. Analyze the effect of changing the confidence level on the width of the confidence interval.
2. Analyze the effect of changing the sample size on the width of the confidence interval.
3. Analyze the effect of changing the standard deviation on the width of the confidence interval.

STEP 3

a. Changing from a $95\%$ confidence level to a $90\%$ confidence level affects the width of the confidence interval. A lower confidence level corresponds to a smaller t-value, which results in a narrower confidence interval.
Conclusion for part (a): B. The interval will become narrower.

STEP 4

b. Changing from a sample size of 400 to a sample size of 30 affects the width of the confidence interval. A smaller sample size increases the standard error $\frac{s}{\sqrt{n}}$ , which results in a wider confidence interval.
Conclusion for part (b): A. The interval will become wider.

STEP 5

c. Changing from a standard deviation of 30 pounds to a standard deviation of 15 pounds affects the width of the confidence interval. A smaller standard deviation decreases the standard error $\frac{s}{\sqrt{n}}$ , which results in a narrower confidence interval.
Conclusion for part (c): B. The interval will become narrower.

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