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A simple random sample of size n = 49 is obtained from a population with u = 89 and o = 21.
(a) Describe the sampling distribution of x.
(b) What is P (x > 94.55) ?
(c) What is P (XS 82.85) ?
(d) What is P (86.3 (a) Choose the correct description of the shape of the sampling distribution of x.
O A. The distribution is skewed right.
B. The distribution is approximately normal.
O C. The distribution is uniform.
D. The distribution is skewed left.
E. The shape of the distribution is unknown.

Answer :

Answer:

a) B. The distribution is approximately normal.

b) 0.0322 = 3.22%

c) 0.0202 = 2.02%

Step-by-step explanation:

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

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean  and standard deviation , the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], 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.

Mean and standard deviation:

[tex]\mu = 89, \sigma = 21[/tex]

Sample of 49:

This means that [tex]n = 49, s = \frac{21}{\sqrt{49}} = 3[/tex]

(a) Describe the sampling distribution of x.

By the Central Limit Theorem, approximately normal, and the correct answer is given by option B.

(b) What is P (x > 94.55) ?

This is 1 subtracted by the p-value of Z when X = 94.55, so:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{94.55 - 89}{3}[/tex]

[tex]Z = 1.85[/tex]

[tex]Z = 1.85[/tex] has a p-value of 0.9678.

1 - 0.9678 = 0.0322

So 0.0322 = 3.22%

Question c:

This is the p-value of Z when X = 82.85. So

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{82.85 - 89}{3}[/tex]

[tex]Z = -2.05[/tex]

[tex]Z = -2.05[/tex] has a p-value of 0.0202.

So

0.0202 = 2.02%

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