Question

A random sample of n= 100 observations is selected from a population with u = 30 and 6 = 21. Approximate the probabilities shown below.a. P(x228) b. P(22.1sxs 26.8)c. P(xs 28.2) d. P(x 2 27.0)Click the icon to view the table of normal curve areas.a. P(x228)(Round to three decimal places as needed.)

Answer 1

Problem Statement

We have been given random sample of 100 observations and we have been asked to find the probabilities of getting certain observed values given the population mean of 30 and a standard deviation of 21.

Method

To solve this question, we need to:

1. Find the z-score of the observations. The formula for calculating the z-score is:

`\begin{gathered} z=(X-\mu)/(\sigma) \\ \text{where,} \\ X=\text{ The observed value} \\ \mu=\text{population mean} \\ \sigma=\text{ standard deviation} \end{gathered}`

2. Convert the z-score to probability using the z-score table.

Implementation

Question A

1. Find the z-score of the observations.:

`\begin{gathered} X\ge28 \\ \mu=30,\sigma=21 \\ z\ge(28-30)/(21) \\ z\ge-(2)/(21) \\ \\ \therefore z\ge-0.0952 \end{gathered}`

2. Convert the z-score to probability using the z-score table.:

Using a z-score calculator, we have the probability to be:

`P(z\ge-0.0952)=0.037938`

This probability is depicted in the drawing below:

If the mean is represented by 0 and the right-hand side of 0 has a probability of 0.5, then the probability of getting greater than or equal to 28, is the addition of the probability 0.037938 gotten above with the 0.5 on the right-hand side of zero.

Thus, the answer to Question A is:

`\begin{gathered} P(X\ge28)=0.037938+0.5=0.537938 \\ \\ \therefore P(X\ge28)\approx0.538\text{ (To 3 decimal places)} \end{gathered}`

Question B: