Question

A population of values has a normal distribution with μ = 149.8 μ=149.8 and σ = 68.2 σ=68.2. You intend to draw a random sample of size n = 186 n=186. Please show your answers as numbers accurate to at least 3 decimal places. Find the probability that a single randomly selected value is less than 148.3. Find the probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.

Answer 1

Answer:

There is a 49.20% probability that a single randomly selected value is less than 148.3.

There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by

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

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

A population of values has a normal distribution with $\mu = 149.8$ and $σ=68.2$.

Find the probability that a single randomly selected value is less than 148.3

This is the pvalue of Z when $X = 148.3$.

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

$Z = \frac{148.3 - 149.8}{68.2}$

$Z = -0.02$

$Z = -0.02$ has a pvalue of 0.4920

There is a 49.20% probability that a single randomly selected value is less than 148.3.

Find the probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.

We want to find the mean of the sample, so we have to find the standard deviation of the population. That is

$s = \frac{68.2}{\sqrt{186}} = 5$

Now, we have to find the pvalue of Z when $X = 148.3$.

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

$Z = \frac{148.3 - 149.8}{5}$

$Z = -0.3$

$Z = -0.3$ has a pvalue of 0.3821

There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.