Question

Newly purchased tires of a particular type are supposed to be filled to a pressure of 30 psi. Let m denote the true average pressure. A test is to be carried out to decide whether m differs from the target value. Determine the P-value for each of the following z test statistic values.

a. 2.10

b. -1.75

c. -.55

d. 1.41

e. -5.3

Answer 1

Concepts and reason

Normal distribution: Normal distribution is a continuous distribution of data that has the bell-shaped curve. The normally distributed random variable x has mean and standard deviation.

Also, the standard normal distribution represents a normal curve with mean 0 and standard deviation 1. Thus, the parameters involved in a normal distribution are mean and standard deviation.

Standardized z-score: The standardized z-score represents the number of standard deviations the data point is away from the mean.

• If the z-score takes positive value when it is above the mean (0).

• If the z-score takes negative value when it is below the mean (0).

Fundamentals

Let , then the standard z-score is found using the formula given below:

Where X denotes the individual raw score, denotes the population mean, and denotes the population standard deviation.

Procedure for finding the z-value is listed below:

1.From the table of standard normal distribution, locate the probability value.

2.Move left until the first column is reached.

3.Move upward until the top row is reached.

4.Locate the probability value, by the intersection of the row and column values gives the area to the left of z.

(a)

The P-value for z test statistics value 2.10 is obtained below:

From the information given, the test statistic value is 2.10 and the test is two-tailed.

Follow the instructions to get the probability distribution plot:

1.Choose Graph > Probability Distribution Plot choose View Probability > OK.

2.From Distribution, choose ‘Normal’ distribution.

3.Enter the Mean as 0 and Standard deviation value as 1.

4.Click the Shaded Area tab.

5.Choose X Value and Both tails for the region of the curve to shade.

6.Enter the X value as 2.10.

7.Click OK.

Follow the above instructions to get the probability distribution plot:

From the output, the P-value is 0.0179 which is the one tailed value.

The P-value for the both tailed is,

(b)

The P-value for z test statistics value –1.75is obtained below:

Follow the instructions to get the probability distribution plot:

1.Choose Graph > Probability Distribution Plot choose View Probability > OK.

2.From Distribution, choose ‘Normal’ distribution.

3.Enter the Mean as 0 and Standard deviation value as 1.

4.Click the Shaded Area tab.

5.Choose X Value and Both tails for the region of the curve to shade.

6.Enter the X value as –1.75.

7.Click OK.

Follow the above instructions to get the probability distribution plot:

From the output, the P-value is 0.0401 which is the one tailed value.

The P-value for the both tailed is,

(c)

The P-value for z test statistics value –0.55 is obtained below:

Follow the instructions to get the probability distribution plot:

1.Choose Graph > Probability Distribution Plot choose View Probability > OK.

2.From Distribution, choose ‘Normal’ distribution.

3.Enter the Mean as 0 and Standard deviation value as 1.

4.Click the Shaded Area tab.

5.Choose X Value and Both tails for the region of the curve to shade.

6.Enter the X value as –0.55.

7.Click OK.

Follow the above instructions to get the probability distribution plot:

From the output, the P-value is 0.2912 which is the one tailed value.

The P-value for the both tailed is,

(d)

The P-value for z test statistics value 1.41 is obtained below:

Follow the instructions to get the probability distribution plot:

1.Choose Graph > Probability Distribution Plot choose View Probability > OK.

2.From Distribution, choose ‘Normal’ distribution.

3.Enter the Mean as 0 and Standard deviation value as 1.

4.Click the Shaded Area tab.

5.Choose X Value and Both tails for the region of the curve to shade.

6.Enter the X value as 1.41.

7.Click OK.

Follow the above instructions to get the probability distribution plot:

From the output, the P-value is 0.0793 which is the one tailed value.

The P-value for the both tailed is,

(e)

The P-value for z test statistics value –5.3 is obtained below:

Follow the instructions to get the probability distribution plot:

1.Choose Graph > Probability Distribution Plot choose View Probability > OK.

2.From Distribution, choose ‘Normal’ distribution.

3.Enter the Mean as 0 and Standard deviation value as 1.

4.Click the Shaded Area tab.

5.Choose X Value and Both tails for the region of the curve to shade.

6.Enter the X value as –5.3.

7.Click OK.

Follow the above instructions to get the probability distribution plot:

From the output, the P-value is 0.0000 which is the one tailed value.

The P-value for the both tailed is,

Ans: Part a

The P-value for z test statistics value 2.10 is 0.0358.

Part b

The P-value for z test statistics value –1.75 is 0.0802.

Part c

The P-value for z test statistics value –0.55 is 0.5824.

Part d

The P-value for z test statistics value 1.41 is 0.1586.

Part e

The P-value for z test statistics value –5.3 is 0.0000.