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A Statistics class from a high school with 4,000 students took a survey of the first 35 students who walked through the front door of the school, and asked how far they traveled to school that day. The class plans to run a one-sample t-test to determine if the average travel distance has increased since last year. The students notice that the sample data are right skewed. Which conditions have been satisfied for the t-test?
I. The sample is from a random sample or randomized experiment.
II. The sampling distribution of sample means is approximately normal.
III. The sample size is small relative to the population.
(A) III
(B) I and II
(C) I and III
(D) II and III
(E) I, II, and III
The correct answer is (D).
The sample was a convenience sample (not random). The sampling distribution is approximately normal due to the large sample size (n>30). The sample (35) is less than 10% of the population (4,000).
An advertiser wants to find convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program. They take a random sample of 100 television viewers and ask them how many commercials they could remember after watching a 30 minute TV program. The appropriate t-test was conducted, which resulted in a P-value of 0.15.
Assuming all conditions were met, which of the following is an appropriate conclusion?
(A) Because the P-value is less than 0.05, at the 5% significance level, there is not convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program.
(B) Because the P-value is greater than 0.05, at the 5% significance level, there is convincing evidence that television viewers remember fewer than 4 commercials, on average, after watching a 30 minute TV program.
(C) Because the P-value is greater than 0.01, at the 1% significance level, there is not convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program.
(D) Because the P-value is greater than 0.01, at the 1% significance level, there is convincing evidence that television viewers remember exactly 4 commercials, on average, after watching a 30 minute TV program.
(E) Because the P-value is less than 0.01, at the 1% significance level, there is convincing evidence that television viewers remember fewer than 4 commercials, on average, after watching a 30 minute TV program.
The correct answer is (C).
If the P-value is greater than significance level, fail to reject the null hypothesis and there is not convincing evidence for the alternative hypothesis.
After completing a statistical analysis of a survey of 40 students, the principal of North High School made the following conclusion: reject the null hypothesis; there is convincing evidence that more than 50% of students support a schedule change to have lunch occur earlier in the day.
Which error could have been committed?
(A) Type I error: Conclude that more than 50% of students want earlier lunch, when 50% or less want earlier lunch.
(B) Type I error: Conclude that more than 50% of students want earlier lunch, when more than 50% want earlier lunch.
(C) Type II error: Fail to reject that 50% of students want earlier lunch, when more than 50% want earlier lunch.
(D) Type II error: Fail to reject that 50% of students want earlier lunch, when 50% or less want earlier lunch.
(E) Type II error: Fail to reject that more than 50% of students want earlier lunch, when 50% or less want earlier lunch.
The correct answer is (A).
Type I error is rejecting the null hypothesis when the null hypothesis is actually true.
The correct answer is (D).
The P-value is a measure of the probability of getting sample data like was found or more extreme, if the null hypothesis is true.
The correct answer is (A).
In a two sample test for two proportions, we use the pooled sample proportion 64909064 as the proportion for calculating standard error.
The correct answer is (D).
This is a random sample from one population, not two samples from two populations, so it is a test of association/independence.
Do stain-polyurethane mixes protect wood as well as stain and polyurethane applied in separate coats? Five types of wood will be used, with two boards of each type of wood. One board of each type of wood (randomly selected) will have the stain-polyurethane mix applied to it, and the other will have stain and polyurethane applied in separate coats. Each board will have water poured on it, and the amount of water retained will be measured.
Which significance test is most appropriate?
(A) A chi-square test for goodness of fit
(B) A one-sample z-test for a proportion
(C) A matched-pair t-test for a mean difference
(D) A two-sample t-test for a difference between two means
(E) A two-sample z-test for a difference between two proportions
The correct answer is (C).
This is a matched pairs experimental design where the statistic is the mean of the differences for each pair.
An analysis of 8 used trucks listed for sale in the 48076 zip code finds that the power model ��(�����^)=3.748−0.1395��(�����)ln(price^)=3.748−0.1395ln(miles), for price (in thousands of dollars) and miles driven (in thousands), is an appropriate model of the relationship.
If a used truck has been driven for 47,000 miles, which of the following is closest to the predicted price for the truck?
(A) $9.46
(B) $24.80
(C) $3,210.00
(D) $9,460.00
(E) $24,800.00
The correct answer is (E).
��(�����^)=3.748−0.1395��(47)ln(price^)=3.748−0.1395ln(47) = 3.21.
�3.21=24.8e3.21=24.8 thousand dollars.
The correct answer is (C).
The degrees of freedom are equal to 10 - 2 = 8, so the critical value is t* = 2.306.
The correct answer is (A).
The appropriate inference procedure in this case is a one-sample t test for a mean. The test statistic for this test is �=19.4−200.25�t=n0.2519.4−20.
The correct answer is (C).
A 21-year old college student submits a photo to the website “Guess My Age”. At a later date he checks back and thousands of users have made guesses about his age.
Of the following, which is the best procedure to investigate whether there is convincing statistical evidence that, on average, he is perceived to be less than 21 years old?
(A) One sample t-test for a mean
(B) One sample z-test for a proportion
(C) Matched-pairs t-test for a mean difference
(D) Two-sample t-test for the difference between two means
(E) A chi-square test of association
The correct answer is (A).
The appropriate procedure in this case is a one-sample t-test for a mean.
Students in a statistics class would like to investigate if more than two-thirds of the Earth is water. To answer this question, students tossed an inflatable globe back and forth. After each catch, the student recorded whether the tip of their pointer finger of their right hand was on water or land. In 50 tosses, their finger was on water 38 times.
Assuming all conditions for inference are met, do the data provide convincing statistical evidence at the significance level of α = 0.05 that more than two-thirds of the Earth is water?
(A) Yes, because the P-value of 0.76 is greater than the significance level of 0.05.
(B) Yes, because the P-value of 0.08 is greater than the significance level of 0.05.
(C) Yes, because the P-value of 0.05 is less than the significance level of 0.08.
(D) No, because the P-value of 0.08 is greater than the significance level of 0.05.
(E) No, because the P-value of 0.03 is less than the significance level of 0.05.
The correct answer is (D).
These data do not provide convincing statistical evidence at the significance level of α = 0.05 that more than two-thirds of the Earth is water because the P-value of 0.08 is greater than the significance level of 0.05. Also, 1-PropZTest(p0: 2/3, x: 38, n: 50, Prop > p0) gives a P-value of 0.0808.
According to the National Association of Colleges and Employers, the mean salary for a new college graduate is $45,327. A small college wants to know if the mean salary of their most recent graduates is greater than $45,327. A random sample of 10 recent graduates from the small college was selected and the mean and standard deviation of the salary for those graduates was found.
With all conditions for inference met, a significance test was conducted and a P-value of 0.045 was obtained. Which of the following statements is the most appropriate conclusion using a significance level of α = 0.05?
(A) There is convincing statistical evidence that the mean salary for all recent college graduates from the small college is greater than $45,327.
(B) There is convincing statistical evidence that the mean salary for the sample of 10 college graduates from the small college is greater than $45,327.
(C) There is not convincing statistical evidence that the mean salary for all recent college graduates from the small college is greater than $45,327.
(D) There is not convincing statistical evidence that the mean salary for the sample of 10 college graduates from the small college is greater than $45,327.
(E) There is not convincing statistical evidence that the graduates from this college will always earn more than $45,327.
The correct answer is (A). Because the P-value of 0.045 < α = 0.05 we have convincing statistical evidence that the mean salary for all recent college graduates from the small college is greater than $45,327.
The correct answer is (E).
The test statistic is t = (13.4862 – 0) / 0.5310 = 25.40.
The correct answer is (B).
The P-value for this test is approximately 0, therefore at the significance level α = 0.05, the data provide convincing evidence that the wheel is not fair.
A credit card company claims that the mean time the customers spend on hold is 3.5 minutes. An employee of this company believes that customers spend more than 3.5 minutes on hold. A random sample of n = 36 calls is selected and the mean time the customers in this sample spent on hold was 5.15 minutes. All conditions for inference were met and the P-value for the appropriate hypothesis test was 0.031.
Which of the following statements is the best interpretation of the P-value?
(A) There is a 0.031 probability that the alternative hypothesis is true.
(B) There is a 0.031 probability that the null hypothesis is true.
(C) If the null hypothesis is true, there is a 0.031 probability of finding convincing evidence that 5.15 minutes is greater than 3.5 minutes.
(D) If the null hypothesis is true, the probability of finding convincing evidence that the null hypothesis is true is 0.031.
(E) If the null hypothesis is true, the probability of observing a sample mean of at least 5.15 minutes is 0.031.
Submit
The correct answer is (E).
The P-value of any significance test gives the probability of obtaining the observed sample result (or farther), given that the null hypothesis is true. In this case, the P-value is the probability of observing a sample mean of at least 5.15 minutes given that the null hypothesis is true.
A machine that makes dimes is adjusted to make them at diameter 1.9 cm. Production records show that when the machine is properly adjusted, it will make dimes with a mean diameter of 1.9 cm and with a standard deviation of 0.1 cm. During production, an inspector checks the diameters of dimes to see if the machine has slipped out of adjustment. A random sample of 64 dimes is selected.
The inspector would like to detect if the true mean diameter happens to reach 1.95 cm at a significance level of a = 0.01. He determines the power of this test to be 0.9228.
What is the probability that the inspector will make a Type II Error?
(A) 0.01
(B) 0.0385
(C) 0.05
(D) 0.0772
(E) 0.9228
The correct answer is (D).
P(Power) + P(Type II Error) = 1, so P(Type II Error) = 1 – P(Power) = 1 – 0.9228 = 0.0772.
An orange juice manufacturer advertises a new orange juice product that contains 50% less sugar. This new product is expected to increase profits substantially because they create the 50% less sugar product by replacing 50% of the juice with water while selling it for the same price. An inspector wants to investigate whether the actual percentage of water in the product is within 5% of the intended 50%.
He selects a random sample of 500 containers of orange juice and determined the percentage of each that is water. He plans to conduct a significance test at the �α = 0.01 level to see if there is convincing evidence that the proportion of water added is different from 0.50.
What is the probability that the inspector makes a Type I error?
(A) 0.01
(B) 0.05
(C) 0.10
(D) 0.45
(E) 0.50
The correct answer is (A).
The probability of making a Type I error is equal to �α = 0.01, the significance level of the test.
