Chapter 6.3, Problem 21E

Future scientists: Education professionals refer to science, technology, engineering, and mathematics as the STEM disciplines. The Alliance for Science and Technology Research in America reported in a recent year that 28% of freshmen entering college planned to major in a STEM discipline. A random sample of 85 freshmen is selected.

1. a. Is it appropriate to use the normal approximation to find the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline? If so, find the probability. If not, explain why not.
2. b. A new sample of 150 freshmen is selected. Find the probability that less than 30% of the freshmen in this sample are planning to major in a STEM discipline.
3. c. Find the probability that the proportion of freshmen in the sample of 150 who plan to major in a STEM discipline is between 0.30 and 0.35.
4. d. Find the probability that more than 32% of the freshmen in the sample of 150 are planning to major in a STEM discipline.
5. e. Would it be unusual if less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline?

To determine

Check whether it is appropriate to use the normal approximation to find the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline if it is appropriate find the probability and if not explain the reason.

Answer to Problem 21E

Yes, it is appropriate to use the normal approximation.

The probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline is 0.6593.

Explanation of Solution

Calculation:

The given information is that 28% of freshmen entering college planned to major in a STEM discipline. A random sample of 85 freshmen is considered.

Central Limit Theorem of proportions:

If $np\ge 10$ and $n\left(1-p\right)\ge 10$ where n is the sample size and p is the population proportion then the sample proportion $\widehat{p}$ is approximately normal with mean ${\mu }_{\widehat{p}}=p$ and standard deviation ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$.

From the given information, the probability of freshmen entering college planned to major in a STEM discipline is 0.28$\left(=\frac{28}{100}\right)$.

Requirement check:

Condition 1: $np\ge 10$

Condition 2: $n\left(1-p\right)\ge 10$

Condition 1: $np\ge 10$

Substitute 85 for n and 0.28 for p in np,

$\begin{array}{c}np=\left(85\right)\left(0.28\right)\\ =23.8\end{array}$

Thus, the requirement $np\left(=23.8\right)\ge 10$ is satisfied.

Condition 2: $n\left(1-p\right)\ge 10$

Substitute 85 for n and 0.28 for p in $n\left(1-p\right)$,

$\begin{array}{c}n\left(1-p\right)=\left(85\right)\left(1-0.28\right)\\ =85\left(0.72\right)\\ =61.2\end{array}$

Thus, the requirement of $n\left(1-p\right)\left(=61.2\right)\ge 10$.

Since the both requirements are satisfied. Thus, it is appropriate to use the normal approximation.

Hence, the normal approximation is appropriate to use for finding the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline.

For mean ${\mu }_{\stackrel{⌢}{p}}$:

Substitute $p=0.28$ in the formula of ${\mu }_{\widehat{p}}$,

${\mu }_{\widehat{p}}=0.28$

Thus, the value of ${\mu }_{\widehat{p}}$ is 0.28.

For standard deviation ${\sigma }_{\widehat{p}}$:

Substitute $n=85$ and $p=0.28$ in the formula of ${\sigma }_{\widehat{p}}$,

$\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{0.28\left(1-0.28\right)}{85}}\\ =\sqrt{\frac{0.28\left(0.72\right)}{85}}\\ =\sqrt{\frac{0.2016}{85}}\\ \approx 0.04870\end{array}$

Thus, the value of ${\sigma }_{\widehat{p}}$ is 0.04870.

The probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline represents the area to the left of 0.30$\left(=\frac{30}{100}\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.28 and Standard deviation as 0.04870.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as 0.30.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline is 0.6593.

To determine

Find the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline.

Answer to Problem 21E

The probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline is 0.7073.

Explanation of Solution

Calculation:

The given information is that about 28% of freshmen entering college planned to major in a STEM discipline in the sample of 150 freshmen.

If $\widehat{p}$ is the sample proportion of a simple random sample of size n and it is drawn from the population with population proportion p then the mean and standard deviation of the sampling distribution of $\widehat{p}$ are ${\mu }_{\widehat{p}}=p$ and ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

Substitute $p=0.28$ in ${\mu }_{\widehat{p}}=p$,

${\mu }_{\widehat{p}}=0.28$

Thus, the mean ${\mu }_{\widehat{p}}$ is 0.28.

The formula for finding standard deviation ${\sigma }_{\widehat{p}}$ is,

${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$,

Substitute $n=150$ and $p=0.28$

$\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{0.28\left(1-0.28\right)}{150}}\\ =\sqrt{\frac{0.28\left(0.72\right)}{150}}\\ =\sqrt{\frac{0.2016}{150}}\\ \approx 0.03666\end{array}$

Thus, the value of ${\sigma }_{\widehat{p}}$ is 0.03666.

The probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline represents the area to the left of 0.30$\left(=\frac{30}{100}\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.28 and Standard deviation as 0.03666.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as 0.30.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that less than 30% of the freshmen in the sample are planning to major in a STEM discipline is 0.7073.

To determine

Find the probability that the proportion of freshmen in the samples of 150 who plan to major in a STEM discipline is between 0.30 and 0.35.

Answer to Problem 21E

The probability that the proportion of freshmen in the samples of 150 who plan to major in a STEM discipline is between 0.30 and 0.35 is 0.2646.

Explanation of Solution

Calculation:

The probability that the proportion of freshmen in the samples of 150 who plan to major in a STEM discipline is between 0.30 and 0.35 represents the area to the right of 0.30 and the area to the left of 0.35.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.28 and Standard deviation as 0.03666.
• Click the Shaded Area tab.
• Choose X value and Middle for the region of the curve to shade.
• Enter the X value 1 as 0.30 and X value 2 as 0.35.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that the proportion of freshmen in the samples of 150 who plan to major in a STEM discipline is between 0.30 and 0.35 is 0.2646.

To determine

Find the probability that more than 32% of the freshmen in the sample of 150 are planning to major in a STEM discipline.

Answer to Problem 21E

The probability that more than 32% of the freshmen in the sample of 150 are planning to major in a STEM discipline is 0.1376.

Explanation of Solution

Calculation:

The probability that more than 32% of the freshmen in the sample of 150 are planning to major in a STEM discipline represents the area to the right of 0.32$\left(=\frac{32}{100}\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.28 and Standard deviation as 0.03666.
• Click the Shaded Area tab.
• Choose X value and Right Tail for the region of the curve to shade.
• Enter the X value as 0.32.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that more than 32% of the freshmen in the sample of 150 are planning to major in a STEM discipline is 0.1376.

To determine

Check whether it is unusual if less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline.

Answer to Problem 21E

No, it is not unusual if less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline.

Explanation of Solution

Calculation:

Unusual:

If the probability of an event is less than 0.05 then the event is called unusual.

The probability of less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline represents the area to the left of 0.25.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.28 and Standard deviation as 0.03666.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as 0.25.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline is 0.2066.

Here, the probability of less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline is greater than 0.05. That is, $0.2066>0.05$. Thus, the event that ‘less than 25% of the freshmen in the sample of 150 were planning to major in a STEM discipline’ is not unusual.