Statistics  200-If A and B are any two events, then P(A AND B)

Subject: Mathematics    / Statistics
Question

1. True or False. Justify for full credit.

(a) If A and B are any two events, then P(A AND B) = P(A) + P(B).

(b) If the variance of a data set is 0, then all the observations in this data set must be identical.

(c) A normal distribution curve is always symmetric to its mean.

(d) When plotted on the same graph, a distribution with a mean of 60 and a standard deviation

of 5 will look more spread out than a distribution with a mean of 40 and standard deviation

of 8.

(e) In a left-tailed test, the value of the test statistic is -2. The test statistic follows a

distribution with the distribution curve shown below. If we know the shaded area is 0.03,

then we have sufficient evidence to reject the null hypothesis at 0.05 level of significance.

2. Choose the best answer. Justify for full credit.

(a) A study was conducted at a local college to analyze the average GPA of students graduated

from UMUC in 2016. 100 students graduated from UMUC in 2016 were randomly selected,

and the average GPA for the group is 3.5. The value 3.5 is a

(i) statistic

(ii) parameter

(iii) cannot be determined

(b) The hotel ratings are usually on a scale from 0 star to 5 stars. The level of this measurement is

(i) interval

(ii) nominal

(iii) ordinal

(iv) ratio

(c) On the day of the last presidential election, UMUC News Club organized an exit poll in which

specific polling stations were randomly selected and all voters were surveyed as they left those

polling stations. This type of sampling is called:

(i) cluster

(ii) convenience

(iii) systematic

(iv) stratified

3. The frequency distribution below shows the distribution for IQ scores for a random sample of

1000 adults. (Show all work. Just the answer, without supporting work, will receive no credit.)

IQ Scores Frequency Cumulative Relative

Frequency

50 – 69 23

70 – 89 249

90 -109 0.722

110 – 129

130 – 149 25

Total 1000

(a) Complete the frequency table with frequency and cumulative relative frequency. Express the

cumulative relative frequency to three decimal places.

(b) What percentage of the adults in this sample has an IQ score of at least 110?

(c) Which of the following IQ score groups does the median of this distribution belong to?

70 – 89, 90 – 109, or 110 – 129? Why?

4. The five-number summary below shows the grade distribution of a STAT 200 quiz for a

sample of 100 students.

Answer each question based on the given information, and explain your answer in each case.

(a) What is the range in the grade distribution?

(b) Which of the following score bands has the most students?

(i) 30 – 50

(ii) 50 – 70

(iii) 70 – 90

(Iv) Cannot be determined

(c) How many students in the sample are in the score band between 70 and 100?

5. Consider selecting one card at a time from a 52-card deck. What is the probability that the first

card is an ace and the second card is also an ace? (Note: There are 4 aces in a deck of cards)

(Show all work. Just the answer, without supporting work, will receive no credit.)

(a) Assuming the card selection is with replacement.

(b) Assuming the card selection is without replacement.

6. There are 2000 students in a high school. Among the 2000 students, 1500 students have a

laptop, and 900 students have a tablet. 500 students have a laptop and a tablet. Let L be the

event that a randomly selected student has a laptop, and T be the event that a randomly selected

student has a tablet. (Show all work. Just the answer, without supporting work, will receive no

credit.)

(a) Provide a written description of the complement event of (L OR T).

(b) What is the probability of complement event of (L OR T)?

7. Consider rolling a fair 6-faced die twice. Let A be the event that the sum of the two rolls is at

most 6, and B be the event that the first one is an even number.

(a) What is the probability that the sum of the two rolls is at most 6 given that the first one is an

even number? Show all work. Just the answer, without supporting work, will receive no credit.

(b) Are event A and event B independent? Explain.

8. Answer the following two questions. (Show all work. Just the answer, without supporting

work, will receive no credit).

(a) The steering committee of UMUC Green Solutions Team consists of 3 committee members. 10

people are interested in serving in the committee. How many different ways can the committee

be selected?

(b) A bike courier needs to make deliveries at 6 different locations. How many different routes can

he take?

9. Let random variable x represent the number of girls in a family of three children.

(a) Construct a table describing the probability distribution. (5 pts)

(b) Determine the mean and standard deviation of x. (Round the answer to two decimal places)

10. Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her

opponent’s serves. Assume her opponent serves 10 times.

(a) Let X be the number of returns that Mimi gets. As we know, the distribution of X is a binomial

probability distribution. What is the number of trials (n), probability of successes (p) and probability of

failures (q), respectively?

(b) Find the probability that that she returns at least 1 of the 10 serves from her opponent.

11. A research concludes that the number of hours of exercise per week for adults is normally

distributed with a mean of 4 hours and a standard deviation of 3 hours. Show all work. Just the

answer, without supporting work, will receive no credit.

(a) Find the 75th percentile for the distribution of exercise time per week. (round the answer to 2 decimal

places)

(b) What is the probability that a randomly selected adult has more than 7 hours of exercise per week?

(round the answer to 4 decimal places)

12. Based on the performance of all individuals who tested between July 1, 2012 and June 30, 2015,

the GRE Verbal Reasoning scores are normally distributed with a mean of 150 and a standard

deviation of 8.45. (https://www.ets.org/s/gre/pdf/gre_guide_table1a.pdf). Show all work. Just

the answer, without supporting work, will receive no credit.

(a) Consider all random samples of 36 test scores. What is the standard deviation of the sample

means?

(b) What is the probability that 36 randomly selected test scores will have a mean test score that is

between 148 and 152?

13. An insurance company checks police records on 500 randomly selected auto accidents and notes

that teenagers were at the wheel in 80 of them. Construct a 95% confidence interval estimate of

the proportion of auto accidents that involve teenage drivers. Show all work. Just the answer,

without supporting work, will receive no credit.

14. A city built a new parking garage in a business district. For a random sample of 60 days, daily

fees collected averaged $2,000, with a standard deviation of $400. Construct a 95% confidence

interval estimate of the mean daily income this parking garage generates. Show all work. Just

the answer, without supporting work, will receive no credit.

15. ABC Company claims that the proportion of its employees investing in individual investment

accounts is higher than national proportion of 45%. A survey of 200 employees in ABC

Company indicated that 100 of them have invested in an individual investment account.

Assume we want to use a 0.10 significance level to test the claim.

(a) Identify the null hypothesis and the alternative hypothesis.

(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting

work, will receive no credit.

(c) Determine the P-value for this test. Show all work; writing the correct P-value, without

supporting work, will receive no credit.

(d) Is there sufficient evidence to support ABC Company’s claim that the proportion of its employees

investing in individual investment accounts is higher than 45%? Explain.

16. Mimi was curious if regular excise really helps weight loss, hence she decided to perform a

hypothesis test. A random sample of 5 UMUC students was chosen. The students took a 30-

minute exercise every day for 6 months. The weight was recorded for each individual before

and after the exercise regimen. Does the data below suggest that the regular exercise helps

weight loss? Assume Mimi wants to use a 0.05 significance level to test the claim.

Weight (pounds)

Subject Before After

1 150 130

2 170 160

3 185 180

4 160 160

5 200 180

(a) Identify the null hypothesis and the alternative hypothesis.

(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting

work, will receive no credit.

(c) Determine the p-value. Show all work; writing the correct critical value, without supporting

work, will receive no credit.

(d) Is there sufficient evidence to support the claim that regular exercise helps weight loss? Justify

your conclusion.

17. In a pulse rate research, a simple random sample of 100 men results in a mean of 80 beats per

minute, and a standard deviation of 11 beats per minute. Based on the sample results, the

researcher concludes that the pulse rates of men have a standard deviation greater than 10 beats

per minutes. Use a 0.05 significance level to test the researcher’s claim.

(a) Identify the null hypothesis and alternative hypothesis.

(b) Determine the test statistic. Show all work; writing the correct test statistic, without

supporting work, will receive no credit.

(c) Determine the P-value for this test. Show all work; writing the correct P-value, without

supporting work, will receive no credit.

(d) Is there sufficient evidence to support the researcher’s claim? Explain.

18. The UMUC MiniMart sells five different types of teddy bears. The manager reports that the

five types are equally popular. Suppose that a sample of 1000 purchases yields observed

counts 230, 170, 210, 180, and 210 for types 1, 2, 3, 4, and 5, respectively. Use a 0.05

significance level to test the claim that the five types are equally popular. Show all work and

justify your answer.

Type 1 2 3 4 5

Number of Teddy Bears 230 170 210 180 210

(a) Identify the null hypothesis and the alternative hypothesis.

(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting

work, will receive no credit.

(c) Determine the P-value. Show all work; writing the correct P-value, without supporting work,

will receive no credit.

(d) Is there sufficient evidence to support the manager’s claim that the five types of teddy bears

are equally popular? Justify your answer.

19. A STAT 200 instructor believes that the average quiz score is a good predictor of final exam

score. A random sample of 10 students produced the following data where x is the average quiz

score and y is the final exam score.

x 80 93 50 60 100 40 85 70 75 85

y 72 95 75 68 90 35 83 60 77 85

(a) Find an equation of the least squares regression line. Show all work; writing the correct

equation, without supporting work, will receive no credit.

(b) Based on the equation from part (a), what is the predicted final exam score if the average quiz

score is 90? Show all work and justify your answer.

20. A study of 10 different weight loss programs involved 500 subjects. Each of the 10 programs had

50 subjects in it. The subjects were followed for 12 months. Weight change for each subject was

recorded. We want to test the claim that the mean weight loss is the same for the 10 programs.

(a) Complete the following ANOVA table with sum of squares, degrees of freedom, and mean

square (Show all work):

Source of Variation Sum of Squares

(SS)

Degrees of Freedom

(df)

Mean Square

(MS)

Factor

(Between) 27.82

Error

(Within)

Total 985.07 499 N/A

(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting

work, will receive no credit.

(c) Determine the P-value. Show all work; writing the correct P-value, without supporting work,

will receive no credit.

(d) Is there sufficient evidence to support the claim that the mean weight loss is the same for the 10

programs at the significance level of 0.05? Explain.