Questions: The probability that x̄ will be greater than 32 is 0.9953. Part: 1/2 Part 2 of 2 (b) Find the 65th percentile of x̄. Round the answer to at least two decimal places. The 65th percentile of x̄ is .

The probability that x̄ will be greater than 32 is 0.9953.

Part: 1/2

Part 2 of 2
(b) Find the 65th percentile of x̄. Round the answer to at least two decimal places.

The 65th percentile of x̄ is .

Transcript text: The probability that $\bar{x}$ will be greater than 32 is 0.9953 . Part: $1 / 2$ Part 2 of 2 (b) Find the $65^{\text {th }}$ percentile of $\bar{x}$. Round the answer to at least two decimal places. The $65^{\text {th }}$ percentile of $\bar{x}$ is $\square$.

Solution

Solution Steps

To find the 65th percentile of a normally distributed variable $\bar{x}$, we need to determine the z-score that corresponds to the 65th percentile and then use the mean and standard deviation of the distribution to find the actual value. This involves using the inverse of the cumulative distribution function (CDF) for the normal distribution.

Step 1: Determine the z-score for the 65th Percentile

To find the 65th percentile of the variable $ \bar{x} $, we first calculate the z-score corresponding to this percentile. The z-score for the 65th percentile is approximately $ z = 0.3853 $.

Step 2: Use the Mean and Standard Deviation

Assuming the mean $ \mu = 30 $ and the standard deviation $ \sigma = 5 $, we can use the z-score to find the actual value of the 65th percentile using the formula: \[ P_{65} = \mu + z \cdot \sigma \]

Step 3: Calculate the 65th Percentile

Substituting the values into the formula: \[ P_{65} = 30 + 0.3853 \cdot 5 \] Calculating this gives: \[ P_{65} \approx 31.9266 \]