Question

A study found that highway drivers in one state traveled at an average speed of 56.6 miles per hour​ (MPH). Assume the population standard deviation is 5.1 MPH. Complete parts a through d below.
a. What is the probability that a sample of 30 of the drivers will have a sample mean less than 56 ​MPH?
b. What is the probability that a sample of 45 of the drivers will have a sample mean less than 56 ​MPH?
c. What is the probability that a sample of 60 of the drivers will have a sample mean less than 56​ MPH?
d. Explain the difference in these probabilities. (Select one each)

i. As the sample size​ increases, the standard error of the mean
(Same, Increase, Decrease)
ii. As the sample size​ increases, the standard error of the mean
(Same, Move farther, Move closer) the population mean of 57.3 MPH.​
iii. Therefore, the probability of observing a sample mean less than 56 MPH
(Same, Increase, Decrease).

Answer 1

Answer:

a) 25.78% probability that a sample of 30 of the drivers will have a sample mean less than 56 ​MPH

b) 21.48% probability that a sample of 45 of the drivers will have a sample mean less than 56 ​MPH

c) 18.14% probability that a sample of 60 of the drivers will have a sample mean less than 56​ MPH

d)

i. As the sample size​ increases, the standard error of the mean

decreases

ii. As the sample size​ increases, the standard error of the mean

move farther the population mean of 57.3 MPH.​

iii. Therefore, the probability of observing a sample mean less than 56 MPH

decrease.

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem

Normal probability distribution:

Problems of normally distributed samples are 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.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$, the sample means with size n of at least 30 can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 56.6, \sigma = 5.1$

a. What is the probability that a sample of 30 of the drivers will have a sample mean less than 56 ​MPH?

This means that $s = \frac{5.1}{\sqrt{30}} = 0.93$

This probability is the pvalue of Z when X = 56. So

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

By the Central Limit theorem

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

$Z = \frac{56 - 56.6}{0.93}$

$Z = -0.65$

$Z = -0.65$ has a pvalue of 0.2578.

25.78% probability that a sample of 30 of the drivers will have a sample mean less than 56 ​MPH

b. What is the probability that a sample of 45 of the drivers will have a sample mean less than 56 ​MPH?

$s = \frac{5.1}{\sqrt{45}} = 0.76$

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

$Z = \frac{56 - 56.6}{0.76}$

$Z = -0.79$

$Z = -0.79$ has a pvalue of 0.2148.

21.48% probability that a sample of 45 of the drivers will have a sample mean less than 56 ​MPH

c. What is the probability that a sample of 60 of the drivers will have a sample mean less than 56​ MPH?

$s = \frac{5.1}{\sqrt{60}} = 0.66$

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

$Z = \frac{56 - 56.6}{0.66}$

$Z = -0.91$

$Z = -0.91$ has a pvalue of 0.1814.

18.14% probability that a sample of 60 of the drivers will have a sample mean less than 56​ MPH

d. Explain the difference in these probabilities. (Select one each)

n increases, so s decreases(moving farther from the mean), which means that z decreases, leaving 56 mph to a lower percentile. So

i. As the sample size​ increases, the standard error of the mean

decreases

ii. As the sample size​ increases, the standard error of the mean

move farther the population mean of 57.3 MPH.​

iii. Therefore, the probability of observing a sample mean less than 56 MPH

decrease.