Question

Suppose a simple random sample of size n= 11 is obtained from a population with u = 62 and a = 14.

(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities regarding the sample me
(b) Assuming the normal model can be used, determine P(x < 65.8).
(c) Assuming the normal model can be used, determine P(x 2 64.2).
Click here to view the standard normal distribution table (page 1).
Click here to view the standard normal distribution table (page 2).
(a) What must be true regarding the distribution of the population?
O A. Since the sample size is large enough, the population distribution does not
need to be normal.
B. The population must be normally distributed and the sample size must be large.
OC. The population must be normally distributed.
OD. There are no requirements on the shape of the distribution of the population.

Answer 1

a) C. The population must be normally distributed.

b) P(x < 65.8) = 0.8159

c) P(x > 64.2) = 0.3015

Explanation:

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

Normal probability distribution

When the distribution is normal, we use the z-score formula.

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

`Z = (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 normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question:

`\mu = 62, \sigma = 14, n = 11, s = (14)/(√(11)) = 4.22`

(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities regarding the sample me

n < 30, so the distribution of the population must be normal.

The correct answer is:

C. The population must be normally distributed.

(b) Assuming the normal model can be used, determine P(x < 65.8).

This is the pvalue of Z when X = 65.8. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (65.8 - 62)/(4.22)`

`Z = 0.9`

`Z = 0.9` has a pvalue of 0.8159.

So

P(x < 65.8) = 0.8159

(c) Assuming the normal model can be used, determine P(x > 64.2).

This is 1 subtracted by the pvalue of Z when X = 64.2. So

`Z = (X - \mu)/(s)`

`Z = (64.2 - 62)/(4.22)`

`Z = 0.52`

`Z = 0.52` has a pvalue of 0.6985.

1 - 0.6985 = 0.3015

So

P(x > 64.2) = 0.3015