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

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
From the given information, the probability of freshmen entering college planned to major in a STEM discipline is 0.28
Requirement check:
Condition 1:
Condition 2:
Condition 1:
Substitute 85 for n and 0.28 for p in np,
Thus, the requirement
Condition 2:
Substitute 85 for n and 0.28 for p in
Thus, the requirement of
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
Substitute
Thus, the value of
For standard deviation
Substitute
Thus, the value of
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
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.
b.

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
Substitute
Thus, the mean
The formula for finding standard deviation
Substitute
Thus, the value of
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
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.
c.

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.
d.

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
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.
e.

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,
Want to see more full solutions like this?
Chapter 6 Solutions
Essential Statistics
- Question: A company launches two different marketing campaigns to promote the same product in two different regions. After one month, the company collects the sales data (in units sold) from both regions to compare the effectiveness of the campaigns. The company wants to determine whether there is a significant difference in the mean sales between the two regions. Perform a two sample T-test You can provide your answer by inserting a text box and the answer must include: Null hypothesis, Alternative hypothesis, Show answer (output table/summary table), and Conclusion based on the P value. (2 points = 0.5 x 4 Answers) Each of these is worth 0.5 points. However, showing the calculation is must. If calculation is missing, the whole answer won't get any credit.arrow_forwardBinomial Prob. Question: A new teaching method claims to improve student engagement. A survey reveals that 60% of students find this method engaging. If 15 students are randomly selected, what is the probability that: a) Exactly 9 students find the method engaging?b) At least 7 students find the method engaging? (2 points = 1 x 2 answers) Provide answers in the yellow cellsarrow_forwardIn a survey of 2273 adults, 739 say they believe in UFOS. Construct a 95% confidence interval for the population proportion of adults who believe in UFOs. A 95% confidence interval for the population proportion is ( ☐, ☐ ). (Round to three decimal places as needed.)arrow_forward
- Find the minimum sample size n needed to estimate μ for the given values of c, σ, and E. C=0.98, σ 6.7, and E = 2 Assume that a preliminary sample has at least 30 members. n = (Round up to the nearest whole number.)arrow_forwardIn a survey of 2193 adults in a recent year, 1233 say they have made a New Year's resolution. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. The 90% confidence interval for the population proportion p is (Round to three decimal places as needed.) J.D) .arrow_forwardLet p be the population proportion for the following condition. Find the point estimates for p and q. In a survey of 1143 adults from country A, 317 said that they were not confident that the food they eat in country A is safe. The point estimate for p, p, is (Round to three decimal places as needed.) ...arrow_forward
- (c) Because logistic regression predicts probabilities of outcomes, observations used to build a logistic regression model need not be independent. A. false: all observations must be independent B. true C. false: only observations with the same outcome need to be independent I ANSWERED: A. false: all observations must be independent. (This was marked wrong but I have no idea why. Isn't this a basic assumption of logistic regression)arrow_forwardBusiness discussarrow_forwardSpam filters are built on principles similar to those used in logistic regression. We fit a probability that each message is spam or not spam. We have several variables for each email. Here are a few: to_multiple=1 if there are multiple recipients, winner=1 if the word 'winner' appears in the subject line, format=1 if the email is poorly formatted, re_subj=1 if "re" appears in the subject line. A logistic model was fit to a dataset with the following output: Estimate SE Z Pr(>|Z|) (Intercept) -0.8161 0.086 -9.4895 0 to_multiple -2.5651 0.3052 -8.4047 0 winner 1.5801 0.3156 5.0067 0 format -0.1528 0.1136 -1.3451 0.1786 re_subj -2.8401 0.363 -7.824 0 (a) Write down the model using the coefficients from the model fit.log_odds(spam) = -0.8161 + -2.5651 + to_multiple + 1.5801 winner + -0.1528 format + -2.8401 re_subj(b) Suppose we have an observation where to_multiple=0, winner=1, format=0, and re_subj=0. What is the predicted probability that this message is spam?…arrow_forward
- Consider an event X comprised of three outcomes whose probabilities are 9/18, 1/18,and 6/18. Compute the probability of the complement of the event. Question content area bottom Part 1 A.1/2 B.2/18 C.16/18 D.16/3arrow_forwardJohn and Mike were offered mints. What is the probability that at least John or Mike would respond favorably? (Hint: Use the classical definition.) Question content area bottom Part 1 A.1/2 B.3/4 C.1/8 D.3/8arrow_forwardThe details of the clock sales at a supermarket for the past 6 weeks are shown in the table below. The time series appears to be relatively stable, without trend, seasonal, or cyclical effects. The simple moving average value of k is set at 2. What is the simple moving average root mean square error? Round to two decimal places. Week Units sold 1 88 2 44 3 54 4 65 5 72 6 85 Question content area bottom Part 1 A. 207.13 B. 20.12 C. 14.39 D. 0.21arrow_forward
- Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGALAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning



