Question

Let \(\bar{X}\) be the mean of a random variable of size 5 from a normal distribution with \(\mu=0\) and variance \(\sigma^2 = 125\). Determine c so that P(\(\bar{X} < c\)) = 0.90
1. 6.10
2. 6.20
3. 6.30
4. 6.40

• A. 6.10
• B. 6.20
• C. 6.30
• D. 6.40

Hint:

The critical \( z \)-scores for common upper-tail areas are very useful to memorize:
- \( 90% \implies z = 1.28 \)
- \( 95% \implies z = 1.645 \)
- \( 99% \implies z = 2.33 \)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires finding a critical value \( c \) for a sample mean distribution, given the population parameters and a target cumulative probability.
Key Formula or Approach:
For a sample of size \( n \) drawn from a normal distribution \( N(\mu, \sigma^2) \), the sample mean \( \bar{X} \) is also normally distributed:
\[ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \] We can standardize \( \bar{X} \) to a standard normal variable \( Z \sim N(0, 1) \) using:
\[ Z = \frac{\bar{X} - \mu_{\bar{X}}}{\sigma_{\bar{X}}} \]

Step 2: Detailed Explanation:
Let us perform the calculations:
- Given Parameters:
Population mean, \( \mu = 0 \)
Population variance, \( \sigma^2 = 125 \)
Sample size, \( n = 5 \)
- Calculate the Mean and Variance of \( \bar{X} \):
\[ \mu_{\bar{X}} = \mu = 0 \] \[ \sigma^2_{\bar{X}} = \frac{\sigma^2}{n} = \frac{125}{5} = 25 \] The standard error (standard deviation of \( \bar{X} \)) is:
\[ \sigma_{\bar{X}} = \sqrt{25} = 5 \] - Standardize the Probability Expression:
We are given:
\[ P(\bar{X} < c) = 0.90 \] Standardizing both sides:
\[ P\left(\frac{\bar{X} - 0}{5} < \frac{c - 0}{5}\right) = 0.90 \] \[ P\left(Z < \frac{c}{5}\right) = 0.90 \] - Find the Critical \( z \)-value:
From the standard normal cumulative distribution table, the value of \( z \) for which \( P(Z < z) = 0.90 \) is approximately \( 1.28 \) (more precisely \( 1.282 \)).
Set the term equal to this \( z \)-score:
\[ \frac{c}{5} = 1.28 \implies c = 5 \times 1.28 = 6.40 \]

Step 3: Final Answer:
The value of \( c \) is \( 6.40 \).
Therefore, the correct choice is Option (D).