Questions: A population of values has a normal distribution with μ=169.2 and σ=50.8. You intend to draw a random sample of size n=214. Find P26, which is the mean separating the bottom 26% means from the top 74% means. P26 (for sample means) = Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or zscores rounded to 3 decimal places are accepted.

Solution Steps

To find \( P_{26} \), which is the mean separating the bottom 26% of sample means from the top 74% of sample means, we need to use the properties of the normal distribution and the concept of the sampling distribution of the sample mean.

1. Find the z-score corresponding to the 26th percentile of the standard normal distribution.
2. Calculate the standard error of the mean using the population standard deviation and the sample size.
3. Convert the z-score to the corresponding sample mean using the population mean and the standard error.

To find the z-score corresponding to the 26th percentile of the standard normal distribution, we use the inverse cumulative distribution function (CDF). The z-score for the 26th percentile is approximately: \[ z = -0.6433 \]

The standard error of the mean is calculated using the population standard deviation \( \sigma \) and the sample size \( n \): \[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{50.8}{\sqrt{214}} \approx 3.4726 \]

Using the z-score, the population mean \( \mu \), and the standard error, we calculate the sample mean \( P_{26} \): \[ P_{26} = \mu + z \times \text{Standard Error} = 169.2 + (-0.6433) \times 3.4726 \approx 166.9659 \]

Final Answer