A manager wishes to see if the time (in minutes) it takes for their workers to complete a certain task is different if they are wearing earbuds.   A random sample of 20 workers' times were collected before and after wearing earbuds. Test the claim that the time to complete the task will be different, i.e. meaning has production differed at all, at a significance level of α = 0.01

For the context of this problem, μ_D = μ_before−μ_after where the first data set represents before earbuds and the second data set represents the after earbuds. Assume the population is normally distributed. The hypotheses are:

H₀: μ_D = 0
H₁: μ_D  0

Before

After

69

65.3

69.5

61.6

39.3

21.4

66.7

60.4

38.3

46.9

85.9

76.6

70.3

77.1

59.8

51.3

72.1

69

79

83

61.7

58.8

55.9

44.7

56.8

50.6

71

63.4

80.6

68.9

59.8

35.5

73.1

77

49.9

38.4

56.2

55.4

64.3

55.6

In a game of cards you win $1 if you draw an even number, $2 if you draw diamonds, $5 if you draw a king and $10 if you draw the queen of diamonds. You get nothing for any other card that you draw.

1. In any random draw, how much money should you expect to win?

2. What is the standard deviation of expected winnings?

The frequency distribution shown in the following table lists the number of hours per day a randomly selected sample of teenagers spent watching television. Where possible, determine what percent of the teenagers spent the following number of hours watching television. (Round your answers to one decimal place. If not possible, enter IMPOSSIBLE.)

(a) less than 4 hours
%

(b) at least 5 hours
%

(c) at least 1 hour
%

(d) less than 2 hours
%

(e) at least 2 hours but less than 4 hours
%

(f) more than 3.5 hours
%

More than 100 million people around the world are not getting enough sleep; the average adult needs between 7.5 and 8 hours of sleep per night. College students are particularly at risk of not getting enough shut-eye. A recent survey of several thousand college students indicated that the total hours of sleep time per night, denoted by the random variable X, can be approximated by a normal model with E(X) = 6.79 hours and SD(X) = 1.22 hours. Question 1. Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x between 6.62 and 6.95. .0039 Incorrect: Your answer is incorrect. (use 4 decimal places in your answer) Question 2. Find the probability that the hours of sleep per night for a random sample of 16 college students has a mean x between 6.62 and 6.95. (use 4 decimal places in your answer) Question 3. Find the probability that the hours of sleep per night for a random sample of 25 college students has a mean x between 6.62 and 6.95. (use 4 decimal places in your answer) Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.

Problem #8

Among 300 employees in a company 100 had Engineering degree, 110 had MBA degree, and 70 had both Engineering and MBA degrees. If an employee from this company is selected at random, find the probability that the employee

a. Has Engineering degree but not MBA..

b. Engineering or MBA degree.

Power +, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows a normal distribution with a mean of 36 hours and a standard deviation of 5.3 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries. a. What can you say about the shape of the distribution of the sample mean? Sample mean Round answers to four decimal places. b. What is the standard error of the distribution of the sample mean? Note: The standard error is the standard deviation of the distribution of the sample mean. That is, the standard error is the population standard deviation divided by the square root of the sample size. Standard error 0.7667 c. What proportion of the samples will have a mean useful life of more than 37 hours? Probability d. What is the probability that a randomly selected AA battery will have a useful life of at most 34 hours? Probability e. What proportion of the samples will have a mean useful life between 34 and 37 hours? Probability

5. A study was performed to compare the e↵ect of dietary supplementation with a low-fibre refined wheat product on the serum cholesterol of 20 healthy participants aged 23-49 years. Each subject had a cholesterol level measured at baseline and then was randomly assigned to receive either a high-fibre or lowfibre diet for 6 weeks. A 2-week period followed during which no supplements were taken. Participants then took the alternative supplement for a 6-week period. The results are shown below:

n    Baseline    High fiber low fiber difference(high fibre - low fibre) difference(high fibre - baseline)    difference(low fibre-baseline)

total Cholesterol(mg/dl)    20 186 +-31    172+-28 172+-25 -1    -14    -13

(-8, +7) (-21,7)    (-20,7)
(a) Does the high-fibre diet change cholesterol level from baseline? Justify your answer.

(b)Does the low-fibre diet change cholesterol level from baseline? Justify your answer.

(c)Is the change due to the high-fibre diet di↵erent from the change due to the low-fibre diet? Justify.

The reference desk of a university library receives requests for assistance. Assume that a Poisson probability distribution with an arrival rate of 11 requests per hour can be used to describe the arrival pattern and that service times follow an exponential probability distribution with a service rate of 15 requests per hour. (Round your answers to four decimal places.)

(a)

What is the probability that no requests for assistance are in the system?

(b)

What is the average number of requests that will be waiting for service?

(c)

What is the average waiting time (in hours) before service begins?

h

(d)

What is the average time (in hours) at the reference desk (waiting time plus service time)?

  h

(e)

What is the probability that a new arrival has to wait for service?

The average age of students at INTI International University is known to be 22.5 years old. Assume the population is normally distributed with a standard deviation of 5 years. Find the probability that the mean age of a sample of 64 students is between 20.9 and 23.8 years old.

Q11. Use the given information to find the number of degrees of​ freedom, the critical values

chi Subscript Upper L Superscript 2χ2L

and

chi Subscript Upper R Superscript 2χ2R​,

and the confidence interval estimate of

sigmaσ.

It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution.

Platelet Counts of Women

90​%

​confidence;

n=40​,

s=65.5.

The confidence interval estimate of sigmaσ is:

_____less than<sigmaσless than<_____

​(Round to one decimal place as​ needed.)

A researcher at a college hears students complain that they don’t have enough time to in their week to study. He believes that the students at the college are spending much more time on Facebook, Twitter, and Instagram than they did three years ago. He knows that three years ago, the mean number of hours per week students spent of social media was 15.1 hours. He takes a sample of 16 students and finds they spend 23.3 hours per week on social media with SS=240.

a)Conduct a t-test to see is his theory is correct. (Be sure to show all 4 steps of a hypothesis test. Also be sure to consider if this is a one-tailed or two-tailed test.)

b)Measure effect size using Cohen's D.

c)Measure effect size using percentage of variance explained (r).

d)Estimate effect size by constructing 90% an confidence interval.

Three products compete in the same market. In 2019, market shares were distributed as follows: Product A: 50% Product B: 15% Product C: 35%
1000 people were asked which of the three products would buy today. The answers were distributed as follows: Company A:480 Company B:120 Company C:400
Use an appropriate hypothesis test to test if the distribution today is significantly different from the distribution in 2019. Use a 5% significance level.

5) (1 point) Triathlon times. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30 - 34 group while Mary competed in the Women, Ages 25 - 29 group. Leo completed the race in 4534 seconds, while Mary completed the race in 5534 seconds. Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Round all calculated answers to four decimal places.

Here is some information on the performance of their groups:

The finishing times of the Men, Ages 30 - 34 group has a mean of 4315 seconds with a standard deviation of 599 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5274 seconds with a standard deviation of 786 seconds.

The distributions of finishing times for both groups are approximately Normal.

Remember: a better performance corresponds to a faster finish.

1. Write the short-hand for these two normal distributions.

The Men, Ages 30 - 34 group has a distribution of N( ,  ).

The Women, Ages 25 - 29 group has a distribution of N( ,  ).

2. What is the Z score for Leo's finishing time? z =

3. What is the Z score for Mary's finishing time? z =

4. Did Leo or Mary rank better in their respective groups?

A. Mary ranked better
B. Leo ranked better
C. They ranked the same

5. What percent of the triathletes did Leo finish slower than in his group?  %.

6. What percent of the triathletes did Mary finish faster than in her group?  %.

7. What is the cutoff time for the fastest 16% of athletes in the men's group, i.e. those who took the shortest 16% of time to finish?

seconds

8. What is the cutoff time for the slowest 31% of athletes in the women's group?

seconds