Question

Suppose that we will randomly select a sample of 69 measurements from a population having a mean equal to 22 and a standard deviation equal to 8.
(a) Describe the shape of the sampling distribution of the sample mean . Do we need to make any assumptions about the shape of the population? Why or why not?
(b) Find the mean and the standard deviation of the sampling distribution of the sample mean . (Round your σx¯ answer to 1 decimal place.)
(c) Calculate the probability that we will obtain a sample mean greater than 24; that is, calculate P( > 24). Hint: Find the z value corresponding to 24 by using µ and σ because we wish to calculate a probability about . (Use the rounded standard error to compute the rounded Z-score used to find the probability. Round your answer to 4 decimal places. Round z-scores to 2 decimal places.).
(d) Calculate the probability that we will obtain a sample mean less than 20.253; that is, calculate P( < 20.253) (Use the rounded standard error to compute the rounded Z-score used to find the probability. Round your answer to 4 decimal places. Round z-scores to 2 decimal places.) Next Visit question mapQuestion 1 of 15 Total1 of 15 PrevMcGraw Hill Education.

Answer 1

Answer:

a) By the Central Limit Theorem, it is going to be bell-shaped, that is, normally distributed. The Central Limit Theorem states that no matter the shape of the population, the shape of the sampling distribution of the sample mean is going to be normally distributed, so we do not need to make any assumptions about the shape of the population.

b) $\mu = 22, s = 0.9630$

c) There is an 1.88% probability that we will obtain a sample mean greater than 24.

d) There is a 6.30% probability that we will obtain a sample mean less than 20.253.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean \mu and standard deviation $\sigma$, a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.

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:

There are 69 measurements, so $n = 69$

The population has a mean of 22, so $\mu = 22$

The population has a standard deviation of 8, so $\sigma = 8$.

(a) Describe the shape of the sampling distribution of the sample mean . Do we need to make any assumptions about the shape of the population? Why or why not?

By the Central Limit Theorem, it is going to be bell-shaped, that is, normally distributed. The Central Limit Theorem states that no matter the shape of the population, the shape of the sampling distribution of the sample mean is going to be normally distributed, so we do not need to make any assumptions about the shape of the population.

(b) Find the mean and the standard deviation of the sampling distribution of the sample mean.

The mean of the sampling distribution of the sample mean is the same as the mean of the population, so $\mu = 22$

The standard deviation of the sampling distribution of the sample mean is the standard deviation of the population divided by the length of the sample. So

$s = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{69}} = 0.9630$

(c) Calculate the probability that we will obtain a sample mean greater than 24

We are working with the sample mean, so we use $s$ in the place of $\sigma$.

This probability is 1 subtracted by the pvalue of Z when $X = 24$.

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

$Z = \frac{24 - 22}{0.9630}$

$Z = 2.08$

$Z = 2.08$ has a pvalue of 0.9812.

This means that there is a 1-0.9812 = 0.0188 = 1.88% probability that we will obtain a sample mean greater than 24.

(d) Calculate the probability that we will obtain a sample mean less than 20.253

This is the pvalue of Z when $X = 20.53$. So:

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

$Z = \frac{20.53 - 22}{0.9630}$

$Z = -1.53$

$Z = -1.53$ has a pvalue of 0.0630.

This means that there is a 6.30% probability that we will obtain a sample mean less than 20.253.