Thensive Quiz
In a sample of 10 randomly selected women, it was found that their mean height was 63.4 inches. From previous studies, it can be assumed that the population standard
deviation is 2.4 and that the population of height measurements is normally distributed. Construct the confidence interval for the population mean. 12 possible
A.
B.
C.
D.

Ask by Lang Estrada. in the United States

Dec 13,2024

Question

Thensive Quiz
In a sample of 10 randomly selected women, it was found that their mean height was 63.4 inches. From previous studies, it can be assumed that the population standard
deviation is 2.4 and that the population of height measurements is normally distributed. Construct the confidence interval for the population mean. 12 possible
A.
B.
C.
D.

Ask by Lang Estrada. in the United States

Dec 13,2024

UpStudy AI · Accepted Answer

To construct a $$ 95\% $$ confidence interval for the population mean, we can use the formula for the confidence interval when the population standard deviation is known:

$$
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
$$

Where:
- $$ \bar{x} $$ is the sample mean
- $$ z $$ is the z-score corresponding to the desired confidence level
- $$ \sigma $$ is the population standard deviation
- $$ n $$ is the sample size

Given:
- Sample mean $$ \bar{x} = 63.4 $$ inches
- Population standard deviation $$ \sigma = 2.4 $$
- Sample size $$ n = 10 $$

1. **Find the z-score for a $$ 95\% $$ confidence level**:
   The z-score for a $$ 95\% $$ confidence level is approximately $$ 1.96 $$.

2. **Calculate the standard error (SE)**:
   $$
   SE = \frac{\sigma}{\sqrt{n}} = \frac{2.4}{\sqrt{10}} \approx \frac{2.4}{3.1623} \approx 0.758
   $$

3. **Calculate the margin of error (ME)**:
   $$
   ME = z \cdot SE = 1.96 \cdot 0.758 \approx 1.484
   $$

4. **Construct the confidence interval**:
   $$
   \text{Lower limit} = \bar{x} - ME = 63.4 - 1.484 \approx 61.916
   $$
   $$
   \text{Upper limit} = \bar{x} + ME = 63.4 + 1.484 \approx 64.884
   $$

Thus, the $$ 95\% $$ confidence interval for the population mean is approximately $$ (61.9, 64.9) $$.

The correct answer is:
**A. $$ (61.9, 64.9) $$**.
The $$ 95\% $$ confidence interval for the population mean is approximately $$ (61.9, 64.9) $$.

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