Question

1. The SAT test scores have an average value of 1200 with a standard deviation of 105. A random sample of 35 scores is selected for study. A) What is the shape, mean(expected value) and standard deviation of the sampling distribution of the sample mean for samples of size 35? B) What is the probability that the sample mean will be larger than 1235? C) What is the probability that the sample mean will fall within 25 points of the population mean? D) What is the probability that the sample mean will be less than 1175?

Answer 1

Answer:

a) The shape is bell shaped, because of the single peak at the center, that is the mean.

The mean of the sampling distribution of the sample mean is 1200.

The standard deviation of the sampling distribution of the sample mean for samples of size 35 is 17.75.

b) There is a 2.07% probability that the sample mean will be larger than 1235.

c) 85.68% probability that the sample mean will fall within 25 points of the population mean.

d) There is a 7.22% probability that the sample mean will be less than 1175.

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

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

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 This p-value is the probability that the value of the measure is greater than X.

In this case, we have that:

The SAT test scores have an average value of 1200 with a standard deviation of 105. A random sample of 35 scores is selected for study. So $\mu = 1200, \sigma = 105, n = 35$.

A) What is the shape, mean(expected value) and standard deviation of the sampling distribution of the sample mean for samples of size 35?

The shape is bell shaped, because of the single peak at the center, that is the mean.

The mean of the sampling distribution of the sample mean is the same as the sample mean. So, the mean of the sampling distribution of the sample mean is 1200.

The standard deviation of the sampling distribution of the sample mean for samples of size 35 is the sample standard deviation divided by the square root of the size of the sampling distribution.

$s = \frac{105}{sqrt{35}} = 17.75$

The standard deviation of the sampling distribution of the sample mean for samples of size 35 is 17.75.

B) What is the probability that the sample mean will be larger than 1235?

This is 1 subtracted by the pvalue of Z when $X = 1235$.

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

$Z = \frac{1235 - 1200}{17.15}$

$Z = 2.04$

$Z = 2.04$ has a pvalue of 0.9793

So there is a 1-0.9793 = 0.0207 = 2.07% probability that the sample mean will be larger than 1235.

C) What is the probability that the sample mean will fall within 25 points of the population mean?

This is the subtraction of the pvalue of the Z score when X = 1225 by the pvalue of the Z score when X = 1175.

So:

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

$Z = \frac{1225 - 1200}{17.15}$

$Z = 1.46$

$Z = 1.46$ has a pvalue of 0.9279

X = 1175 is going to have $Z = -1.46$, that has a pvalue of 0.0722.

This means that there is a 0.9297 - 0.0729 = 0.8568 = 85.68% probability that the sample mean will fall within 25 points of the population mean.

D) What is the probability that the sample mean will be less than 1175?

X = 1175 has $Z = -1.46$, that has a pvalue of 0.0722.

This means that there is a 7.22% probability that the sample mean will be less than 1175.