# Distributions, Hypothesis Testing, and Linear Regression

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An introductory statistics question pack from the Sofia Open Content Initiative.

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Assume the amount of money seventh–grade students spend on food each day at school is exponentially distributed with an average of $2.50. Which graph best describes the distribution:

A

B

C

D

Assume the amount of money seventh–grade students spend on food each day at school is exponentially distributed with an average of $2.50. Find the probability that a randomly selected seventh–grade student spends less than $4.00 a day on food.

0.7981

0.2019

0.9999

0.0001

Assume the amount of money seventh–grade students spend on food each day at school is exponentially distributed with an average of $2.50. 85% of the seventh–grade students spend more than what amount per day ?

$ 2.12

$ 0.75

$ 4.74

$ 0.41

The amount of time that intermediate algebra students at Leland High School spend doing their homework per day is normally distributed with a mean 1.5 hours and standard deviation 0.75 hours. If one student is randomly chosen, what is the probability that the student does intermediate algebra homework at least 2 hours per day?

0.7475

0.4259

0.2525

0.6784

The amount of time that intermediate algebra students at Leland High School spend doing their homework per day is normally distributed with a mean 1.5 hours and standard deviation 0.75 hours. 60% of these students spend at most how many hours doing their homework?

1.69 hours

1.31 hours

1.5 hours

0.2533 hours

Llamas are excellent pack animals. It is known that the weight of supplies carried by llamas follows a normal distribution with a mean of 62.5 pounds and a standard deviation of 6 pounds. Find the probability that the weight of supplies carried by one randomly chosen llama is between 60 and 70 pounds?

0.4441

0.5559

0.8944

1

Llamas are excellent pack animals. It is known that the weight of supplies carried by llamas follows a normal distribution with a mean of 62.5 pounds and a standard deviation of 6 pounds. The middle 50% of weights of supplies carried by a randomly chosen llama is between ________ and ________.

0 and 62.5 pounds

58.45 and 66.55 pounds

56.5 and 68.5 pounds

There is not enough information given.

Which of the following are true for the normal distribution?

I More values fall close to the mean than fall far away from the mean. II The mean and standard deviation cannot be the same. III A change in µ causes the graph to shift to the left or right and changes the shape of the graph. IV A change in s causes a change in the shape of the normal curve.

I , IV

I , II , III , IV

I , II , III

III , IV

The length of time junior high school students sleep per night follows an approximate uniform distribution from seven to eleven hours. Suppose we randomly select a junior high student. Find the probability that the randomly selected student sleeps less than 8 1/2 hours per night.

0.2143

0.7727

0.4705

0.3750

The length of time junior high school students sleep per night follows an approximate uniform distribution from seven to eleven hours. Suppose we randomly select a junior high student. Find the probability that the randomly selected student sleeps eight to twelve hours per night.

0

1.0

0.7500

0.25

The length of time junior high school students sleep per night follows an approximate uniform distribution from seven to eleven hours. Suppose we randomly select a junior high student. On average, how long does a junior high school student sleep per night ?

4 hours

9 hours

8 hours

11 hours

The length of time junior high school students sleep per night follows an approximate uniform distribution from seven to eleven hours. Suppose we randomly select a junior high student. 65% of junior high school students sleep at least how many hours ?

9.6 hours

6.5 hours

7.8 hours

8.4 hours

The length of time junior high school students sleep per night follows an approximate uniform distribution from seven to eleven hours. Suppose we randomly select a junior high student. We are interested in the probability that a randomly selected student sleeps less than eight hours, knowing that he/she sleeps less than ten. Which graph best depicts this situation?

A

B

C

D

Last year, President Bush granted a tax-cut check to all income tax filers. In doing so, it was reported that he thought that at least 30% of the households would use the tax-cut check to increase spending. According to a report by the University of Michigan Research Center (Wall Street Journal, Feb. 26, 2002), 220 of the 1000 people surveyed said that the 2001 tax-cut check they received has led them to increase spending. The alternate hypothesis for this test is:

p > 0.30

p > 0.22

p < 0.30

p < 0.22

Last year, President Bush granted a tax-cut check to all income tax filers. In doing so, it was reported that he thought that at least 30% of the households would use the tax-cut check to increase spending. According to a report by the University of Michigan Research Center (Wall Street Journal, Feb. 26, 2002), 220 of the 1000 people surveyed said that the 2001 tax-cut check they received has led them to increase spending. A Type II error is to:

Conclude that the proportion of people that would use the tax-cut check to increase spending is less than 30% when in fact the proportion is at least 30%.

Conclude that the proportion of people that would use the tax-cut check to increase spending is at least 30% when in fact the proportion is less than 30%.

Conclude that the proportion of people that would use the tax-cut check to increase spending is less than 22% when in fact the proportion is at least 22%.

Conclude that the proportion of people that would use the tax-cut check to increase spending is at least 22% when in fact the proportion is less than 22%.

Last year, President Bush granted a tax-cut check to all income tax filers. In doing so, it was reported that he thought that at least 30% of the households would use the tax-cut check to increase spending. According to a report by the University of Michigan Research Center (Wall Street Journal, Feb. 26, 2002), 220 of the 1000 people surveyed said that the 2001 tax-cut check they received has led them to increase spending. Which of the following is the correct decision to make for the test?

Reject the null hypothesis.

Do not reject the null hypothesis.

The test is inconclusive.

According to an article by George Will (San Jose Mercury News, Feb. 28, 2002), the average U.S. consumption per person per year of French Fries is 28 pounds. Suppose that you believe that the average in Santa Clara County is not 28 pounds. You randomly survey 50 people in this county. The sample average is 24 pounds with a sample standard deviation of 10 pounds. Conduct an appropriate hypothesis test. This test is $\rule{1cm}{0.15mm}$

left-tailed

right-tailed

two-tailed

no-tailed

According to an article by George Will (San Jose Mercury News, Feb. 28, 2002), the average U.S. consumption per person per year of French Fries is 28 pounds. Suppose that you believe that the average in Santa Clara County is not 28 pounds. You randomly survey 50 people in this county. The sample average is 24 pounds with a sample standard deviation of 10 pounds. Conduct an appropriate hypothesis test. The p-value for this test is $\rule{1cm}{0.15mm}$

0.0068

0.0034

0.0047

0.0136

According to an article by George Will (San Jose Mercury News, Feb. 28, 2002), the average U.S. consumption per person per year of French Fries is 28 pounds. Suppose that you believe that the average in Santa Clara County is not 28 pounds. You randomly survey 50 people in this county. The sample average is 24 pounds with a sample standard deviation of 10 pounds. Conduct an appropriate hypothesis test. At the 5% level, the correct conclusion is:

The average consumption in Santa Clara County is less than 28 pounds.

The average consumption in Santa Clara County is not 28 pounds.

The average consumption in Santa Clara County is less than 24 pounds.

The average consumption in Santa Clara County is 24 pounds.

A hospital administrator wants to determine the proportion of emergency room patients that use the emergency room (ER) for non-emergency care. She randomly samples records from 350 ER patients and determines that 74 of those patients required only non-emergency care. The administrator constructs a 90% confidence interval for the true proportion of all ER patients who use the ER for non-emergency care. The error bound for the proportion EBP for this confidence interval is:

0.036

0.072

0.030

0.106

A hospital administrator wants to determine the proportion of emergency room patients that use the emergency room (ER) for non-emergency care. She randomly samples records from 350 ER patients and determines that 74 of those patients required only non-emergency care. If the same data were used but the confidence level used was 95% instead of 90%, the error bound for the proportion EBP would be:

larger

the same

smaller

we are unable to determine unless another sample is obtained

What is meant by the term “95% confident” when constructing a confidence interval for a mean?

If we took repeated samples, approximately 95% of the samples would produce the same confidence interval.

If we took repeated samples, approximately 95% of the confidence intervals calculated from those samples would contain the sample mean.

If we took repeated samples, the sample mean would equal the population mean in approximately 95% of the samples.

If we took repeated samples, approximately 95% of the confidence intervals calculated from those samples would contain the true value of the population mean.

An aircraft manufacturer is testing a new procedure to be used in installing a certain component in an aircraft. For a random sample of 8 airplanes being assembled, the time (in minutes) required to install the component for each of these 8 aircrafts are:

80 84 87 91 91 95 102 106

Assume the underlying population of installation times is approximately normally distributed. Find the 90% confidence interval for the true mean installation time using this new procedure.

(86.1, 97.9)

(86.9, 91.1)

(87.2, 96.8)

Not enough information

An aircraft manufacturer is testing a new procedure to be used in installing a certain component in an aircraft. For a random sample of 8 airplanes being assembled, the time (in minutes) required to install the component for each of these 8 aircrafts are:

80 84 87 91 91 95 102 106

Assume the underlying population of installation times is approximately normally distributed. The value that is the center of the confidence interval is $\rule{1cm}{0.15mm}$.

91

92

93

µ

The amount of soda contained in a can for a certain brand of soda is normally distributed with a population standard deviation of 0.1 ounces. A random sample of 40 cans of soda was selected and the amount of soda in each can was measured. The sample mean was 12.03 ounces and the sample standard deviation was 0.08 ounces. What is the appropriate distribution to use when calculating a confidence interval for the true mean amount of soda contained in all cans of this brand?

The student-t distribution, because the sample standard deviation is given.

The student-t distribution, because the repair costs are approximately normally distributed.

The standard normal distribution, because the population standard deviation is known.

The standard normal distribution, because the sample mean is known.

Two competing parcel-delivery firms in a large city make conflicting claims about which one delivers parcels in the shortest time. A random sample of 100 delivery times for the first company produces a sample mean of x1 = 37 minutes and a sample standard deviation of s1 = 10 minutes. A random sample of 100 delivery times for the second company produces a sample mean of x2 = 41 minutes and a sample standard deviation of s2 = 12 minutes. Conduct a hypothesis test to determine if the mean delivery time for the first company is less than that for the second company. The null hypothesis is:

m1 < m2

m1 > m2

m1 £ m2

m1 ³ m2

Two competing parcel-delivery firms in a large city make conflicting claims about which one delivers parcels in the shortest time. A random sample of 100 delivery times for the first company produces a sample mean of x1 = 37 minutes and a sample standard deviation of s1 = 10 minutes. A random sample of 100 delivery times for the second company produces a sample mean of x2 = 41 minutes and a sample standard deviation of s2 = 12 minutes. Conduct a hypothesis test to determine if the mean delivery time for the first company is less than that for the second company. The exact distribution for the hypothesis test is:

The normal distribution because the population standard deviations are given.

The student-t distribution because the population standard deviations are unknown.

The exponential distribution because the time of delivery decreases.

Not able to determine.

Two competing parcel-delivery firms in a large city make conflicting claims about which one delivers parcels in the shortest time. A random sample of 100 delivery times for the first company produces a sample mean of x1 = 37 minutes and a sample standard deviation of s1 = 10 minutes. A random sample of 100 delivery times for the second company produces a sample mean of x2 = 41 minutes and a sample standard deviation of s2 = 12 minutes. Conduct a hypothesis test to determine if the mean delivery time for the first company is less than that for the second company. If the p-value is 0.0056, the conclusion is:

The mean delivery time for the first company is higher than the mean delivery time for the second company.

The mean delivery time for the first company is no more than the mean delivery time for the second company.

The mean delivery time for the first company is at least equal to the mean delivery time for the second company.

The mean delivery time for the first company is less than the mean delivery time for the second company.

Participants in a random sample of 10 professional football players are placed on a yogurt-and-banana diet for one month. The weights before and after one month on the diet are as follow:

Before: 187 205 165 193 199 286 212 189 242 253 After: 175 193 167 190 197 240 210 189 221 255

We want to determine if the yogurt-and-banana diet helps reduce the weight of the football players. Assume the weight of the professional football players is approximately normally distributed.

This is a test of:

Two independent population means, population standard deviations known.

Two independent population means, population standard deviations unknown.

Paired or matched samples.

Two population proportions.

Participants in a random sample of 10 professional football players are placed on a yogurt-and-banana diet for one month. The weights before and after one month on the diet are as follow:

Before: 187 205 165 193 199 286 212 189 242 253 After: 175 193 167 190 197 240 210 189 221 255

We want to determine if the yogurt-and-banana diet helps reduce the weight of the football players. Assume the weight of the professional football players is approximately normally distributed.

The distribution for the hypothesis test is $\rule{1cm}{0.15mm}$

Student-t

Exponential

Normal

Uniform

A newspaper/TV network survey was conducted to determine whether the percentage of adult males who favor the death penalty is greater than the percentage of adult females who favor the death penalty. A random sample of 800 adult males produced 480 who favor the death penalty., while a random sample of 800 adult females produced 410 who favor the death penalty. The Type I error is to:

Conclude that the percentage of adult males who favor the death penalty is greater than the percentage of adult females who favor the death penalty when, in fact, the percentage of males is no more than the percentage of females.

Conclude that the percentage of adult males who favor the death penalty is no more than the percentage of adult females who favor the death penalty when, in fact, the percentage of males is more than the percentage of females.

Conclude that the percentage of adult males who favor the death penalty is greater than the percentage of adult females who favor the death penalty when, in fact, the percentage of males is actually greater than the percentage of females.

Conclude that the percentage of adult males who favor the death penalty is no more than the percentage of adult females who favor the death penalty when, in fact, the percentage of males is actually no more than the percentage of females.

New snowboarders (those who have snowboarded a year or less) often suffer from minor injuries. A random sample of seven new snowboarders produced the data on number of months snowboarding and number of minor injuries in the last month that they snowboarded. Is the correlation between the number of months snowboarding and the number of injuries in the last month snowboarding significant?

Yes

No

Not enough information to answer question

New snowboarders (those who have snowboarded a year or less) often suffer from minor injuries. A random sample of seven new snowboarders produced the data on number of months snowboarding and number of minor injuries in the last month that they snowboarded. The linear regression equation is:

$\hat{y}$ = 9.5904 – 0.9614 x

$\hat{y}$ = -0.7349 + 9.5904 x

$\hat{y}$ = 9.5904 + 0.7349 x

$\hat{y}$ = 9.5904 – 0.7349 x

New snowboarders (those who have snowboarded a year or less) often suffer from minor injuries. A random sample of seven new snowboarders produced the data on number of months snowboarding and number of minor injuries in the last month that they snowboarded. If a new snowboarder has snowboarded for five (5) months, how many injuries would s/he have in the last month snowboarding?

4.8

47.2

5.9

13.3

If you calculate the line of best fit and the independent variable and the dependent variable have negative correlation, then the line of best fit has slope zero (0).

True

False

There is not enough information

The cost of a leading gourmet ice cream in different sizes is given in the table. Are there any outliers?

Yes

No

Not enough information

The cost of a leading gourmet ice cream in different sizes is given in the table. If your friend used the line of best fit to predict the cost for a 128-ounce size of gourmet ice cream, what would you tell him/her with what you have learned about linear regression?

That a line is not the best fit for the cost of the gourmet ice cream.

That the 128-ounce size of the gourmet ice cream is far too expensive.

That we should switch the independent and dependent variables.

That s/he should not use the line of best fit to make this prediction.

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