Question

In a certain normal distribution, find the mean μ when σ = 5 and 5.48% of the area lies to the left of 78.

Answer 1

The mean of the normal distribution, using the formula for the z-score, considering the given data, is of 86.

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

In this problem:

• The standard deviation is of $\sigma = 5$.

• 5.48% of the area lies to the left of 78, hence when X = 78, Z has a p-value of 0.0548, that is, Z = -1.6.

Then:

$Z = \frac{X - \mu}{\sigma}$

$-1.6 = \frac{78 - \mu}{5}$

$78 - \mu = -1.6(5)$

$\mu = 86$

More can be learned about the normal distribution at https://brainly.com/question/24663213