Week 7 Homework

A student wants to determine if there is a difference in the pricing between two stores for health and beauty supplies. She recorded prices from both stores for each of 10 different products. Assuming that the conditions for conducting the test are​ satisfied, determine if there is a price difference between the two stores. Use the α=0.005 level of significance. ​(a) What type of test should be​ used? (b) Determine the null and alternative hypotheses. ​(c) Use technology to calculate the​ P-value. (d) Draw a conclusion based on the hypothesis test.

(a) A hypothesis test regarding the difference of two means using a​ matched-pairs design. (b) H₀: µd=0 and H₁: µd≠0 (c) STAT-->EDIT-->Store 1 in L1, Store 2 in L2, (L1-L2) in L3 STAT-->TESTS-->T-Test (Inpt=Data, µ₀=0, List=L3, Freq=1, ≠µ₀) p=0.331 (d) There is not sufficient evidence to reject the null hypothesis because the P-value>α.

Do women feel differently from men when it comes to tax​ rates? One question on a survey of randomly selected adults​ asked, "What percent of income do you believe individuals should pay in income​ tax?" ​(a) Explain why a hypothesis test may be used to test whether the mean tax rates for the two genders differ. (b) Test whether the mean tax rate for females differs from that of males at the α=0.1 level of significance. Determine the null and alternative hypotheses for this test. Let μM represent the mean income tax rate for males and let μF represent the mean income tax rate for females. (c) Find t₀​, the test statistic for this hypothesis test, and the P-value. (d) State the appropriate conclusion.

(a) Each sample is obtained independently of the other. Each sample is a simple random sample. Each sample size is small relative to the size of its population. Each sample size is large. (b) H₀: µM=µF and H₁: µM≠µF (c) STAT-->EDIT-->Males into L1, Females into L2, (L1-L2) into L3 STAT-->TESTS-->2-SampTTest (Inpt=Data, List1=L2, List2=L1, Freq1=1, Freq2=1, ≠µ₀, Pooled=No t₀ = -0.99 p = 0.324 (d) Do not reject H₀. There is not sufficient evidence at the level of significance to conclude that the mean income tax rate for males is different from the mean income tax rate for females.

The manufacturer of a certain engine treatment claims that if you add their product to your​ engine, it will be protected from excessive wear. An infomercial claims that a woman drove 3 hours without​ oil, thanks to the engine treatment. A magazine tested engines in which they added the treatment to the motor​ oil, ran the​ engines, drained the​ oil, and then determined the time until the engines seized. ​(a) Determine the null and alternative hypotheses that the magazine will test. (b) Both engines took exactly 19 minutes to seize. What conclusion might the magazine make based on this​ evidence?

(a) H₀: µ = 3 H₁: µ < 3 (b) The​ infomercial's claim is not true.

Assume that both populations are normally distributed. ​a) Test whether μ1≠μ2 at the α=0.01 level of significance for the given sample data. (b) Determine the test statistic. (c) Approximate the​ P-value. Choose the correct answer below. (d) Should the hypothesis be rejected at the α=0.01 level of significance​? ​b) Construct a 99​% confidence interval about μ1−μ2.

(a) H₀: µ₁=µ₂ and H₁: µ₁≠µ₂ (b) STAT-->TESTS-->2-SampTTest (Inpt=Stats, x1=14.7, Sx1=3.9, n1=12, x2=15.8, Sx2=4.6, n2=12, ≠µ₂, Pooled=no) t₀=-0.63 (c) p=0.534, so P-value≥0.10 (d) Do not reject the null hypothesis because the ​P-value is greater than or equal to the level of significance. (e) STAT-->TESTS-->2-SampTInt (Inpt=Stats, x1=14.7, Sx1=3.9, n1=12, x2=15.8, Sx2=4.6, n2=12, C-Level=0.99, Pooled=no) The confidence interval is the range from −6.02 to 3.82.

Several years​ ago, 42​% of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive. A recent poll asked 1,005 parents who have children in grades​ K-12 if they were satisfied with the quality of education the students receive. Of the 1,005 surveyed, 464 indicated that they were satisfied. Construct a 90​% confidence interval to assess whether this represents evidence that​ parents' attitudes toward the quality of education have changed. (a) What are the null and alternative​ hypotheses? (b) Use technology to find the 90​% confidence interval. (c) What is the correct​ conclusion?

(a) H₀​: p=0.42 versus H₁​: p≠0.42 (b) STAT-->TESTS-->1-PropZInt (464, 1005, 0.9) The lower bound is 0.44. The upper bound is 0.49. (c) Since the interval does not contain the proportion stated in the null​ hypothesis, there is sufficient evidence that​ parents' attitudes toward the quality of education have changed.

In 1945​, an organization asked 1477 randomly sampled American​ citizens, "Do you think we can develop a way to protect ourselves from atomic bombs in case others tried to use them against​ us?" with 761 responding yes. Did a majority of the citizens feel the country could develop a way to protect itself from atomic bombs in 1945​? Use the α=0.01 level of significance. (a) What are the null and alternative​ hypotheses? (b) Determine the test​ statistic, z₀, and the P-value. (c) What is the correct conclusion at the α=0.01 level of​ significance?

(a) H₀​: p=0.50 and H₁​: p>0.50 (b) STAT-->TESTS-->1-PropZTest (p₀ = 0.50, x = 761, n = 1477, >p₀) z₀ = 1.17 p = 0.121 (c) Since the​ P-value is greater than the level of significance, do not reject the null hypothesis. There is not sufficient evidence to conclude that the majority of the citizens feel the country could develop a way to protect itself from atomic bombs.

A math teacher claims that she has developed a review course that increases the scores of students on the math portion of a college entrance exam. Based on data from the administrator of the​ exam, scores are normally distributed with μ=516. The teacher obtains a random sample of 1800 students, puts them through the review​ class, and finds that the mean math score of the 1800 students is 521 with a standard deviation of 114. ​(a) State the null and alternative hypotheses. Let μ be the mean score. (b) Test the hypothesis at the α=0.10 level of significance. Is a mean math score of 521 statistically significantly higher than 516​? Conduct a hypothesis test using the​ P-value approach. ​(c) Do you think that a mean math score of 521 versus 516 will affect the decision of a school admissions​ administrator? In other​ words, does the increase in the score have any practical​ significance? ​(d) Test the hypothesis at the α=0.10 level of significance with n=350 students. Assume that the sample mean is still 521 and the sample standard deviation is still 114. Is a sample mean of 521 significantly more than 516​? Conduct a hypothesis test using the​ P-value approach.

(a) H₀​: μ=516 and H₁​: μ>516 (b) Find the test statistic and P-value: STAT-->TESTS-->T-Test (Stat, μ=516, overbar x = 521, Sx=114, n=1800, >μ₀) t₀ = 1.86 P = 0.031 Is the sample mean statistically significantly​ higher? Yes, because the​ P-value is 0.031​, which is less than α=0.10. (If the​ P-value is less than the level of​ significance, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis.) (c) No, because the score became only 0.97​% greater. (d) Find the test statistic and P-value: t₀ = 0.82 P = 0.206 Is the sample mean statistically significantly​ higher? No because the​ P-value is 0.206​, which is greater than α=0.10. If the​ P-value is less than the level of​ significance, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis. What do you conclude about the impact of large samples on the​ P-value? As n​ increases, the likelihood of rejecting the null hypothesis increases.​ However, large samples tend to overemphasize practically insignificant differences.

A blind taste test is conducted to determine which of two​ colas, Brand A or Brand​ B, individuals prefer. Individuals are randomly asked to drink one of the two types of cola​ first, followed by the other​ cola, and then asked to disclose the drink they prefer. Results of the taste test indicate that 52 of 100 individuals prefer Brand A. (a) Conduct a hypothesis test​ (preferably using​ technology) H₀​: p=p₀ versus H₁​: p≠p₀ for p₀=0.41​, 0.42​, 0.43​, ..., 0.61​, 0.62, 0.63 at the α=0.05 level of significance. For which values of p₀ do you not reject the null​ hypothesis? What do each of the values of p₀ represent? ​(b) Construct a​ 95% confidence interval for the proportion of individuals who prefer Brand A. ​(c) Suppose you changed the level of significance in conducting the hypothesis test to α=0.01. What would happen to the range of values for p₀ for which the null hypothesis is not​ rejected? Why does this make​ sense? Choose the correct answer below.

(a) STAT-->TESTS-->1-PropZTest (0.41, 52, 100, p≠0)-->p=0.025-->Continue trying each p₀ value until you find the lower and upper limits at which p₀>0.05. Do not reject the null hypothesis for the values of p₀ between 0.43 and 0.61​, inclusively. (b) STAT-->TESTS-->1-PropZInt (52, 100, 0.95) The lower bound is 0.422. The upper bound is 0.618. (c) STAT-->TESTS-->1-PropZInt (52, 100, 0.99) = 0.391, 0.649) The range of values would increase because the corresponding confidence interval would increase in size.

A college entrance exam company determined that a score of 21 on the mathematics portion of the exam suggests that a student is ready for​ college-level mathematics. To achieve this​ goal, the company recommends that students take a core curriculum of math courses in high school. Suppose a random sample of 200 students who completed this core set of courses results in a mean math score of 21.7 on the college entrance exam with a standard deviation of 3.6. Do these results suggest that students who complete the core curriculum are ready for​ college-level mathematics? That​ is, are they scoring above 21 on the math portion of the​ exam? (a) State the appropriate null and alternative hypotheses. ​(b) Verify that the requirements to perform the test using the​ t-distribution are satisfied. Select all that apply. (c) Use the​ P-value approach at the α=0.05 level of significance to test the hypotheses in part​ (a). Identify the test statistic and approximate the P-value. (d) Write a conclusion based on the results.

(a) The appropriate null and alternative hypotheses are H₀​: μ=21 versus H₁​: μ>21. (b) The​ students' test scores were independent of one another. The students were randomly sampled. The sample size is larger than 30. (c) STAT-->TESTS-->T-Test (Stats, 21, 21.7, 13.6, 200, >µ₀) t₀ = 2.75 p = 0.003. The P-value is in the range P-value<0.1 (d) Reject the null hypothesis and claim that there is sufficient evidence to conclude that the population mean is greater than 21.

To test the belief that sons are taller than their​ fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their​ fathers? Use the α=0.10 level of significance.​ Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (a) Which conditions must be met by the sample for this​ test? Select all that apply. (b) Write the hypotheses for the test. Use the difference ​"Fathers−​Sons." (c) Calculate the test statistic and P-value. (d) Approximate the​ P-value for this hypothesis test. (e) Should the null hypothesis be​ rejected?

(a) The differences are normally distributed or the sample size is large. The sampling method results in a dependent sample. The sample size is no more than​ 5% of the population size. (b) H₀: µd=0 and H₁: µd>0 (c) STAT-->EDIT-->Xi into L1, Yi into L2, (L1-L2) into L3 STAT-->TESTS-->2-SampTTest (Inpt=Data, List1=L1, List2=L2, Freq1=1, Freq2=1, <µ₂, Pooled=No) t₀ = 0.01 (d) p=0.503 The​ P-value is in the range 0.25 ≤ P-value < 1. (e) Do not reject H₀ because the​ P-value is greater than the level of significance. There is not sufficient evidence to conclude that sons are taller than their fathers at the 0.10 level of significance.

Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H₀​: p=0.2 versus H₁​: p>0.2 n=100​ x=30​ α= 0.1 (a) Is np₀(1−p₀)≥​10? (b) Use technology to find the​ P-value. (c) _______ the null ​hypothesis, because the ​P-value is _______ than α.

(a) Yes. np₀(1-p₀) = (100 * 0.2)*(1-0.2) = 16 (b) STAT→TESTS→1-PropZTest (0.2, 30, 100, >p₀, Calculate) = 0.006 (c) Reject, less

In​ randomized, double-blind clinical trials of a new​ vaccine, monkeys were randomly divided into two groups. Subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. After the second​ dose, 114 of 724 subjects in the experimental group​ (group 1) experienced fever as a side effect. After the second​ dose, 71 of 605 of the subjects in the control group​ (group 2) experienced fever as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the α=0.05 level of​ significance? (a) Verify the model requirements. Select all that apply. (b) Determine the null and alternative hypotheses. (c) Find the test statistic and P-value for this hypothesis test. (d) Interpret the​ P-value. (e) State the conclusion for this hypothesis test.

(a) n₁p-hat₁(1−p₁)≥10 and n₂p-hat₂(1−p₂)≥10 The samples are independent. The sample size is less than​ 5% of the population size for each sample. (b) H₀: p₁ = p₂ and H₁: p₁ > p₂ (c) STAT-->TESTS-->2-PropZTest (x1=114, n1=724, x2=71, n2=605, >p2) z = 2.10 p=0.018 (d) If the population proportions are equal, one would expect a sample difference proportion greater than the one observed in about 18 out of 1000 repetitions of this experiment. (e) Reject H₀. There is sufficient evidence to conclude that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the α=0.05 level of significance.

To test H₀: μ=107 versus H₁: μ≠107 a simple random sample of size n=35 is obtained. (a) Does the population have to be normally distributed to test this​ hypothesis? Why? (b) If overbar x=103.9 and s=5.8​, compute the test statistic. (c) Draw a​ t-distribution with the area that represents the​ P-value shaded. Choose the correct graph below. (d) Approximate the​ P-value. (e) Interpret the​ P-value. (f) If the researcher decides to test this hypothesis at the α=0.01 level of​ significance, will the researcher reject the null​ hypothesis?

(a) ​No, because n≥30. (b) STAT-->TESTS-->T-Test (Stats, 107, 103.9, 5.8, 35, ≠µ₀, Calculate) t₀ = -3.16 (c) Graph with two tails shaded. (d) STAT-->TESTS-->T-Test (Stats, 107, 103.9, 5.8, 35, ≠µ₀, Calculate) p = 0.003, so 0.002<​P-value<0.005 (e) If 1000 random samples of size n=35 are​ obtained, about 3 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=107. (f) Yes, because the​ P-value is less than the level of significance (α=0.05​).

What three conditions must be satisfied before testing a hypothesis regarding population proportion p?

1. The sample is obtained by simple random sampling or the data result from a randomized experiment. 2. The sample comes from a population that is normally distributed with no outliers, or the sample, n, is large (n ≥ 30). 3. The sampled values are independent of each other. This means that the sample size is no more than 5% of the population size (n≤0.05N).

What are the criteria to test the difference between two population proportions (independent sample)?

1. The samples are independently obtained using simple random sampling or the data result from a completely randomized experiment with two levels of treatment. 2. n₁p-hat₁(1 - p-hat₁) ≥10 and n₂p-hat₂(1 - p-hat₂) ≥10 3. The sampled values are independent of each other. This means that each sample size is no more than 5% of the population size (n₁ ≤ 0.05N₁ and n₂ ≤ 0.05N₂). This ensures the independence necessary for a binomial experiment.

What are the five steps for testing a hypothesis about a population mean, µ?

1. Two-tailed is H₀: µ = µ₀ and H₁: µ ≠ µ₀ Left-tailed is H₀: µ = µ₀ and H₁: µ < µ₀ Right-tailed is H₀: µ = µ₀ and H₁: µ > µ₀ 2. Select a level of significance, α, depending on the seriousness of making a Type I Error. 3. Use P-value and test statistic (1 population-mean) on calculator to obtain the P-value: a. STAT-->Edit-->Type data into L₁ (Freq: 1) b. Check for normality and outliers. c. STAT-->TEST-->T-Test Inpt: Stats if you have mean and standard deviation. µ₀ = null hypothesis mean overbar x = sample data mean Sx = sample standard deviation n = sample size µ: ≠, <, > 4. If p-value < α, reject the null hypothesis. If p-value > α, do not reject. 5. State the conclusion.

What are the 5 steps for testing a hypothesis regarding the difference between two population means (dependent samples)?

1. Two-tailed is H₀: µ = µ₀ and H₁: µ ≠ µ₀ Left-tailed is H₀: µ = µ₀ and H₁: µ < µ₀ Right-tailed is H₀: µ = µ₀ and H₁: µ > µ₀ 2. Select a level of significance, α, depending on the seriousness of making a Type I Error. 3. Use P-value and test statistic (2 populations-mean) on calculator to obtain the P-value: a. STAT-->EDIT-->Type data into L1 and L2.TEST-->2-PropZTest (x1: # of successes, n1: # of sample in first proportion, x2: # of successes in second proportion, n2: # of sample in second proportion, p1: ≠, <, >) 4. If p-value < α, reject the null hypothesis. If p-value > α, do not reject. 5. State the conclusion.

What are the five steps for testing a hypothesis regarding the difference between two population proportions (independent sample)?

1. Two-tailed is H₀: µ = µ₀ and H₁: µ ≠ µ₀ Left-tailed is H₀: µ = µ₀ and H₁: µ < µ₀ Right-tailed is H₀: µ = µ₀ and H₁: µ > µ₀ 2. Select a level of significance, α, depending on the seriousness of making a Type I Error. 3. Use P-value and test statistic (2 populations-proportion) on calculator to obtain the P-value: a. STAT-->TEST-->2-PropZTest (x1: # of successes, n1: # of sample in first proportion, x2: # of successes in second proportion, n2: # of sample in second proportion, p1: ≠, <, >) 4. If p-value < α, reject the null hypothesis. If p-value > α, do not reject. 5. State the conclusion.

The manufacturer of hardness testing equipment uses​ steel-ball indenters to penetrate metal that is being tested.​ However, the manufacturer thinks it would be better to use a diamond indenter so that all types of metal can be tested. Because of differences between the two types of​ indenters, it is suspected that the two methods will produce different hardness readings. The metal specimens to be tested are large enough so that two indentions can be made.​ Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings. Construct a​ 95% confidence interval to judge whether the two indenters result in different measurements. ​Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (a) Construct a​ 95% confidence interval to judge whether the two indenters result in different​ measurements, where the differences are computed as ​'diamond minus steel​ ball'. (b) State the appropriate conclusion.

STAT-->EDIT-->Steel ball in L1, Diamond in L2, (L2-L1) in L3. STAT-->TESTS-->TInterval (Inpt=Data, List=L3, Freq=1, C-Level=0.95) Lower level: 0.3 Upper level: 3.1 (b) There is sufficient evidence to conclude that the two indenters produce different hardness readings.

A survey​ asked, "How many tattoos do you currently have on your​ body?" Of the 1239 males​ surveyed, 178 responded that they had at least one tattoo. Of the 1010 females​ surveyed, 133 responded that they had at least one tattoo. Construct a 99​% confidence interval to judge whether the proportion of males that have at least one tattoo differs significantly from the proportion of females that have at least one tattoo. Interpret the interval.

STAT-->TESTS-->2-PropZInt (x1=178, n1=1239, x2=133, n2=1010, C-level=0.99) The lower bound is −0.026. The upper bound is 0.050. There is 99​% confidence that the difference of the proportions is in the interval. Conclude that there is insufficient evidence of a significant difference in the proportion of males and females that have at least one tattoo.

What is the formula for estimating p₁-p₂ if prior estimates of p₁ and p₂ are available?

n = [p-hat₁(1 - p-hat₁)+p-hat₂(1 - p-hat₂)] * [(z sub-α/2)/E]²

A researcher with the Department of Education followed a cohort of students who graduated from high school in a certain​ year, monitoring the progress the students made toward completing a​ bachelor's degree. One aspect of his research was to determine whether students who first attended community college took longer to attain a​ bachelor's degree than those who immediately attended and remained at a​ 4-year institution. The data in the table attached below summarize the results of his study. (a) What is the response variable in this​ study? What is the explanatory​ variable? (b) Explain why this study can be analyzed using inference of two sample means. Determine what qualifications are met to perform the hypothesis test about the difference between two means. (c) Does the evidence suggest that community college transfer students take longer to attain a​ bachelor's degree? Use an α=0.01 level of significance. Perform a hypothesis test. Determine the null and alternative hypotheses. (d) Determine the test statistic and P-value. (e) Should the hypothesis be​ rejected? (f) Construct a 95​% confidence interval for μ(community college)−μ(no transfer) to approximate the mean additional time it takes to complete a​ bachelor's degree if you begin in community college. (g) Do the results of parts​ e) and​ f) imply that community college causes you to take extra time to earn a​ bachelor's degree?

(a) The response variable is the time to graduate. The explanatory variable is the use of community college or not. (b) The samples are independent. The samples can be reasonably assumed to be random. The sample sizes are not more than​ 5% of the population. The sample sizes are large​ (both greater than or equal to​ 30). (c) H₀​: μ(community college)=μ(no transfer) and H₁​: μ(community college)>μ(no transfer) (d) STAT-->TESTS-->2-SampTTest (Inpt=Stats, x1=5.44, Sx1=1.131, n1=253, x2=4.47, Sx2=1.006, n2=1122, >µ2, Pooled=No) t₀ = 12.57 p = 0.001 (e) Reject the null hypothesis. The evidence does suggest that community college transfer students take longer to attain a​bachelor's degree at the α=0.01 level of significance. (f) STAT-->TESTS-->2-SampTInt (Inpt=Stats, x1=5.44, Sx1=1.131, n1=253, x2=4.47, Sx2=1.006, n2=1122, C-Level=0.95, Pooled=no) The confidence interval is the range from 0.818 to 1.122. (g) No.

Explain the difference between an independent and dependent sample.

A sample is independent when an individual selected for one sample does not dictate which individual is to be in the second sample. A sample is dependent when an individual selected for one sample dictates which individual is to be in the second sample. Dependent samples are often referred to as​ matched-pairs samples.

What is the formula for estimating p₁-p₂ if prior estimates of p₁ and p₂ are not available?

n = 0.5* [(z sub-α/2)/E]²

A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if he wishes the estimate to be within 5 percentage points with 99​% ​confidence, assuming that: ​(a) He uses the estimates of 22.3​% male and 19.3​% female from a previous​ year? ​(b) He does not use any prior​ estimates?

z sub-α/2 of 99% = 2.575 E = 0.05 (a) n = [p-hat₁(1 - p-hat₁)+p-hat₂(1 - p-hat₂)] * [(z sub-α/2)/E]² = [(0.223)(1-0.223) + (0.193)(1-0.193)] * (2.575/0.05)² = 873 (b) n = 0.5* [(z sub-α/2)/E]² = 0.5 * (2.575/0.05)² = 1327

Twenty years​ ago, 56​% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 242 of 700 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did twenty years​ ago? Use the α=0.05 level of significance. (a) Because np₀(1−p₀)=_______ _____ ​10, the sample size is _______ 5% of the population​ size, and the sample _______, the requirements for testing the hypothesis _______ satisfied. (b) Find the test statistic. (c) Find the​ P-value. (d) Determine the conclusion for this hypothesis test.

(a) 172.5, >, less than, can be reasonably assumed to be random, are (b) STAT-->TESTS-->1-PropZTest (0.56, 252, 700, <p₀, calculate) z₀ = -11.42 (c) STAT-->TESTS-->1-PropZTest (0.56, 252, 700, <p₀, calculate) p = 0.000 (d) Since P-value<α​, reject the null hypothesis and conclude that there is sufficient evidence that parents feel differently today.

A random sample of n1=135 individuals results in x1=40 successes. An independent sample of n2=140 individuals results in x2=60 successes. Does this represent sufficient evidence to conclude that p1<p2 at the α=0.10 level of​ significance? ​(a) What type of test should be​ used? (b) Determine the null and alternative hypotheses. ​(c) Use technology to calculate the​ P-value. ​(d) Draw a conclusion based on the hypothesis test.

(a) A hypothesis test regarding the difference between two population proportions from independent samples. (b) H₀: p₁=p₂ and H₁: p₁<p₂ (c) STATS-->TEST-->2-PropZTest (x1=40, n1=135, x2=60, n2=140, <p2) p=0.011 (c) There is sufficient evidence to reject the null hypothesis because the P-value<α.

The data table represents the measure of a variable before and after a treatment. Does the sample evidence suggest that the treatment is effective in decreasing the value of the response​ variable? Use the α=0.10 level of significance. ​(a) What type of test should be​ used? ​(b) Determine the null and alternative hypotheses. Let μd=μx−μy. ​(c) Use technology to calculate the​ P-value. ​(d) Draw a conclusion based on the hypothesis test.

(a) A hypothesis test regarding the difference of two means using a​ matched-pairs design. (b) H₀: µd=0 and H₁: µd>0 (c) STAT-->EDIT-->Xi in L1, Yi in L2, (L1-L2) in L3. STAT-->TESTS-->T-Test (Inpt=Data, µ₀=0, List=L3, Freq=1, >µ₀) p = 0.190 (d) There is not sufficient evidence to reject the null hypothesis because the P-value>α.

A can of soda is labeled as containing 13 fluid ounces. The quality control manager wants to verify that the filling machine is not over-filling the cans. (a) Determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly. ​(b) The quality control manager obtains a sample of 84 cans and measures the contents. The sample evidence leads the manager to reject the null hypothesis. Write a conclusion for this hypothesis test. ​(c)​ Suppose, in​ fact, the machine is not out of calibration. Has a Type I or Type II error been​ made? ​(d) Management has informed the quality control department that it does not want to shut down the filling machine unless the evidence is overwhelming that the machine is out of calibration. What level of significance would you recommend the quality control manager to​ use? Explain.

(a) H₀: µ = 13 H₁: µ > 13 (b) There is sufficient evidence to conclude that the machine is out of calibration. (c) A Type I error has been made since the sample evidence led the​ quality-control manager to reject the null​ hypothesis, when the null hypothesis is true. (d) The level of significance should be 0.01 because this makes the probability of Type I error small.

​Previously, 5​% of mothers smoked more than 21 cigarettes during their pregnancy. An obstetrician believes that the percentage of mothers who smoke 21 cigarettes or more is less than 5​% today. She randomly selects 145 pregnant mothers and finds that 4 of them smoked 21 or more cigarettes during pregnancy. Test the​ researcher's statement at the α=0.05 level of significance. (a) What are the null and alternative​ hypotheses? (b) Find the​ P-value. (c) Is there sufficient evidence to support the​ obstetrician's statement?

(a) H₀​: p=0.05 versus H₁​: p<0.05 Because np₀(1−p₀)=(145*0.05)(1-0.05)=6.9 <​ 10, the normal model may not be used to approximate the​ P-value. (b) 2nd-->VARS-->binomcdf (145, 0.05, 4) = 0.145 (c) Compare the​ P-value with α. If the​ P-value is less than α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis. No​, do not reject the null hypothesis because the ​P-value is greater than α. There is not sufficient evidence to conclude that the percentage of mothers who smoke 21 or more cigarettes during pregnancy is less than 5​%.

​Previously, 7.2​% of workers had a travel time to work of more than 60 minutes. An urban economist believes that the percentage has increased since then. She randomly selects 95 workers and finds that 8 of them have a travel time to work that is more than 60 minutes. Test the​ economist's belief at the α=0.05 level of significance. (a) What are the null and alternative​ hypotheses? (b) Find the​ P-value. (c) Is there sufficient evidence to support the​ economist's belief?

(a) H₀​: p=0.072 versus H₁​: p>0.072 (b) 1-binomcdf (95, 0.072, [8-1 because p>]) = 0.376 (c) No​, do not reject the null hypothesis. There is not sufficient evidence because the ​P-value is greater than α.

Suppose the mean​ wait-time for a telephone reservation agent at a large airline is 40 seconds. A manager with the airline is concerned that business may be lost due to customers having to wait too long for an agent. To address this​ concern, the manager develops new airline reservation policies that are intended to reduce the amount of time an agent needs to spend with each customer. A random sample of 250 customers results in a sample mean​ wait-time of 39.4 seconds with a standard deviation of 4.3 seconds. Using α=0.05 level of​ significance, do you believe the new policies were effective in reducing wait​ time? Do you think the results have any practical​ significance? (a) Determine the null and alternative hypotheses. (b) Calculate the test statistic and P-value. (c) State the conclusion for the test. (d) State the conclusion in context of the problem. (e) Do you think the results have any practical​ significance?

(a) H₀​: µ=40 and H₁​: µ<40 (b) STAT-->TESTS-->T-Test (µ₀=40, overbar x=39.4, Sx=4.3, n=250, <µ₀) t₀ = -2.21 p = 0.014 (c) Reject H₀ because the​ P-value is less than the α=0.05 level of significance. (d) There is sufficient evidence at the α=0.05 level of significance to conclude that the new policies were effective. (e) ​No, because while there is significant evidence that shows the new policies were effective in lowering the mean​ wait-time of​ customers, the difference between the previous mean​ wait-time and the new mean​ wait-time is not large enough to be considered important.

The average daily volume of a computer stock in 2011 was μ=35.1 million​ shares, according to a reliable source. A stock analyst believes that the stock volume in 2014 is different from the 2011 level. Based on a random sample of 30 trading days in​ 2014, he finds the sample mean to be 30.6 million​ shares, with a standard deviation of s=14.4 million shares. Test the hypotheses by constructing a 95​% confidence interval. ​(a) State the hypotheses for the test. ​(b) Construct a 95​% confidence interval about the sample mean of stocks traded in 2014. (c) Will the researcher reject the null​ hypothesis?

(a) H₀​: μ=35.1 and H₁​: μ≠35.1 (b) STAT-->TESTS-->TInterval (Stats, overbar x = 30.6, Sx = 14.4, n = 30, C-level = 0.95) The lower bound is 25.223 million shares. The upper bound is 35.977 million shares. (c) Do not reject the null hypothesis because μ=35.1 million shares falls in the confidence interval.

A researcher wondered if attainment within six years among students who receive grants as part of their educational funding​ (Group 1) was lower than attainment within six years among students who did not receive grants as part of their educational funding​ (Group 2). Attainment is defined as whether the student earned the degree or certificate that​ he/she set out to earn upon enrollment. ​(a) Is the response variable qualitative or​ quantitative? ​(b) How many groups are being​ compared? ​(c) In part​ (b), we learned that two groups​ (students who receive grants and students who do not receive​ grants) are being compared. In​ addition, the sampling method is independent. State the null and alternative hypotheses for this test.

(a) Qualitative (b) 2 (c) H₀: p₁=p₂ and H₁: p₁<p₂

Assume that the differences are normally distributed. ​(a) Determine di=Xi−Yi for each pair of data. (b) Compute overbar-d and s sub-d. ​(c) Test if μd<0 at the α=0.05 level of significance. What are the correct null and alternative​ hypotheses? (d) What is the P-value? (e) Choose the correct conclusion below. (f) Compute a​ 95% confidence interval about the population mean difference μd.

(a) STAT-->EDIT-->Xi into L1, Yi into L2, (L1-L2) into L3 to calculate di-Xi-Yi (b) STAT-->TESTS-->TTest (Inpt=Data, µ₀=0, List=L3, Freq=1, <µ₀) Overbar-d = overbar-x = -1.575 S sub-d = Sx = 1.769 (c) H₀: µd=0 and H₁: µd<0 (d) p = 0.020 (e) Reject the null hypothesis. There is sufficient evidence that μd<0 at the α=0.05 level of significance. (f) STAT-->TESTS-->TInterval (Inpt=Data, List=L3, Freq=1, C-Level=0.95) The lower bound is −3.05. The upper bound is −0.10.

The research group asked the following question of individuals who earned in excess of​ $100,000 per year and those who earned less than​ $100,000 per​ year: "Do you believe that it is morally wrong for unwed women to have​ children?" Of the 1,205 individuals who earned in excess of​ $100,000 per​ year, 715 said​ yes; of the 1,310 individuals who earned less than​ $100,000 per​ year, 700 said yes. Construct a​ 95% confidence interval to determine if there is a difference in the proportion of individuals who believe it is morally wrong for unwed women to have children. (a) The lower bound is ______ and the upper bound is ______. (b) Because the confidence interval _______ ​0, there is _______ evidence at the α=0.05 level of significance to conclude that there is a difference in the proportions. It seems that the proportion of individuals who earn over​ $100,000 that feel it is morally wrong for unwed women to have children is _______ the proportion of individuals who earn less than​ $100,000 that feel it is morally wrong for unwed women to have children.

(a) STAT-->TESTS-->2-PropZInt (x1=715, n1=1205, x2=700, n2=1310, C-Level=0.95) Lower: 0.020 Upper: 0.098 (b) Does not include, sufficient, greater than

Two researchers conducted a study in which two groups of students were asked to answer 42 trivia questions from a board game. The students in group 1 were asked to spend 5 minutes thinking about what it would mean to be a​ professor, while the students in group 2 were asked to think about soccer hooligans. These pretest thoughts are a form of priming. The 200 students in group 1 had a mean score of 24.1 with a standard deviation of 4.8​, while the 200 students in group 2 had a mean score of 16.1 with a standard deviation of 2.9. (a) Determine the 90​% confidence interval for the difference in​ scores, μ1−μ2. (b) Interpret the interval. (c) What does this say about​ priming?

(a) STAT-->TESTS-->2-SampTInt (Inpt=Stats, x1=24.1, Sx1=4.8, n1=200, x2=16.1, Sx2=2.9, n2=200, C-Level=0.90, Pooled=No) The lower bound is 7.346. The upper bound is 8.654. (b) The researchers are 90​% confident that the difference of the means is in the interval. (c) Since the 90​% confidence interval does not contain​ zero, the results suggest that priming does have an effect on scores.

A random sample of 40 adults with no children under the age of 18 years results in a mean daily leisure time of 5.02 hours, with a standard deviation of 2.39 hours. A random sample of 40 adults with children under the age of 18 results in a mean daily leisure time of 4.41 hours, with a standard deviation of 1.86 hours. Construct and interpret a 90​% confidence interval for the mean difference in leisure time between adults with no children and adults with children μ1−μ2. Let μ1 represent the mean leisure hours of adults with no children under the age of 18 and μ2 represent the mean leisure hours of adults with children under the age of 18. (a) The 90​% confidence interval for μ1−μ2 is the range from ____ hours to ____ hours. (b) What is the interpretation of this confidence​ interval?

(a) STAT-->TESTS-->2-SampTInt (Inpt=Stats, x1=5.02, Sx1=2.39, n1=40, x2=4.41, Sx2=1.86, n2=40, C-Level=0.90, Pooled=No) The 90​% confidence interval for μ1−μ2 is the range from -0.19 hours to 1.41 hours. (b) There is 90​% confidence that the difference of the means is in the interval. Conclude that there is insufficient evidence of a significant difference in the number of leisure hours.

A researcher studies water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The researcher immerses a weighted disk painted black and white and measures the depth​ (in inches) at which it is no longer visible. The collected data is given in the table below. (a) Why is it important to take the measurements on the same​ date? (b) Does the evidence suggest that the clarity of the lake is improving at the α=0.05 level of​ significance? Note that the normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Let di=Xi−Yi. Identify the null and alternative hypotheses. (c) Determine the test statistic and P-value for this hypothesis test. (d) What is your conclusion regarding H₀​?

(a) Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date. (b) H₀: µd=0 and H₁: µd<0 (c) STAT-->ENTER-->Xi into L1, Yi into L2, (L1-L2) into L3. STAT-->TESTS-->T-Test (Inpt=Data, µ₀=0, List=L3, Freq=1, <µ₀) t = -3.50 p = 0.009 (d) Reject H₀. There is sufficient evidence at the α=0.05 level of significance to conclude that the clarity of the lake is improving.

The mean waiting time at the​ drive-through of a​ fast-food restaurant from the time an order is placed to the time the order is received is 84.9 seconds. A manager devises a new​ drive-through system that he believes will decrease wait time. As a​ test, he initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right. (a) Because the sample size is​ small, the manager must verify that the wait time is normally distributed and the sample does not contain any outliers. The normal probability plot is shown below and the sample correlation coefficient is known to be r=0.978. Are the conditions for testing the hypothesis​ satisfied? (b) Is the new system​ effective? Conduct a hypothesis test using the​ P-value approach and a level of significance of α=0.05. i. First determine the appropriate hypotheses. ii. Find the test statistic. iii. Find the​ P-value. (c) Use the α=0.05 level of significance. What can be concluded from the hypothesis​ test?

(a) Yes, the conditions are satisfied. The normal probability plot is linear​ enough, since the correlation coefficient is greater than the critical value. In ​addition, a boxplot does not show any outliers. (b) i. H₀​: μ=84.9 and H₁​: μ<84.9 ii. STAT-->EDIT-->T-Test (Stats, µ₀=84.9, overbar x=79, Sx=15.14, n=10, <µ₀) t₀ = -1.23 iii. STAT-->EDIT-->T-Test (Stats, µ₀=84.9, overbar x=79, Sx=15.14, n=10, <µ₀) p = 0.125 (c) The​ P-value is greater than the level of significance so there is not sufficient evidence to conclude the new system is effective.

If the consequences of making a Type I error are​ severe, would you choose the level of​ significance, α​, to equal​ 0.01, 0.05, or​ 0.10?

0.01 Choose a small α to make it difficult to reject H₀.

Suppose you flip a coin five times. What is the probability of obtaining five tails in a row assuming the coin is​ fair?

0.5⁵ = 0.03125

What are the criteria for testing hypotheses regarding the difference between two population means (dependent samples)?

1. The sample is obtained by simple random sampling or the data result from a matched-pairs design experiment. 2. The sample data are matched pairs (dependent). 3. The differences are normally distributed with no outliers or the sampling size, n, is large (n ≥ 30). 4. The sampled values are independent of each other. That is, the sample size is no more than 5% of the population size (n ≤ 05N).

A researcher wants to show the mean from population 1 is less than the mean from population 2 in​ matched-pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi​, and di=Xi−Yi​, then the null hypothesis is H0​: μd=0 and the alternative hypothesis is H1​: μd ___ 0.

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Some have argued that throwing darts at the stock pages to decide which companies to invest in could be a successful​ stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 100 companies to invest in. After 1​ year, 53 of the companies were considered​ winners; that​ is, they outperformed other companies in the same investment class. To assess whether the​ dart-picking strategy resulted in a majority of​ winners, the researcher tested H₀​: p=0.5 versus H₁​: p>0.5 and obtained a​ P-value of 0.2743. Explain what this​ P-value means and write a conclusion for the researcher.​ (Assume α is 0.1 or​ less.)

About 27 in 100 samples will give a sample proportion as high or higher than the one obtained if the population proportion really is 0.5. Because the​ P-value is​ large, do not reject the null hypothesis. There is not sufficient evidence to conclude that the​ dart-picking strategy resulted in a majority of winners.

Explain what a​ P-value is. What is the criterion for rejecting the null hypothesis using the​ P-value approach?

A​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true. If P-value<α​, reject the null hypothesis.

According to a​ report, the standard deviation of monthly cell phone bills was $6.11 three years ago. A researcher suspects that the standard deviation of monthly cell phone bills is less today. Determine the null and alternative hypotheses.

H₀: σ = $6.11 H₁: σ < $6.11

If we do not reject the null hypothesis when the statement in the alternative hypothesis is​ true, we have made a Type​ _______ error.

II. A Type I error occurs if the null hypothesis is rejected​ when, in​ fact, the null hypothesis is true. A Type II error occurs if the null hypothesis is not rejected​ when, in​ fact, the alternative hypothesis is true.

A sampling method is _______ when an individual selected for one sample does not dictate which individual is to be in the second sample.

Independent

The null and alternative hypotheses are given. Determine whether the hypothesis test is​ left-tailed, right-tailed, or​ two-tailed. What parameter is being​ tested? H₀​: μ=2 H₁​: μ<2

Left-tailed test. The parameter being tested is the population mean (µ).

What is at the​ "heart" of hypothesis testing in​ statistics?

Make an assumption about​ reality, and collect sample evidence to determine whether it contradicts the assumption.

The _______ ​hypothesis, denoted H0​, is a statement to be​ tested, and is a statement of no​ change, no​ effect, or no difference.

Null

Which of the following scenarios could be analyzed using a randomization approach for two sample​ proportions?

Obtaining a random sample of Republicans and an independent random sample of Democrats. Ask each​ sample, "Do you approve or disapprove of the job the president of the United States is​ doing?" Obtaining 300 volunteers suffering from a skin rash and randomly dividing them into two groups. Group 1 receives an experimental drug once a week for 10​ weeks; group 2 receives a placebo once a week for 10 weeks. After the ten​ weeks, it is determined whether the skin rash cleared​ up, or not.

When the results of a hypothesis test are determined to be statistically​ significant, then we​ _______________ the null hypothesis.

Reject

Construct a confidence interval for p₁−p₂ at the given level of confidence. x₁=29​ n₁=254​ x₂=39​ n₂=288​ 90​% confidence

STAT-->TESTS-->2-PropZInt (x1=29, n1=254, x2=39, n2=288, C-level=0.9) The researchers are 90​% confident the difference between the two population​ proportions, p₁−p₂​, is between −0.068 and 0.025.

Explain what​ "statistical significance" means.

Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true.

Explain the difference between statistical significance and practical significance.

Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application.

Assuming all model requirements for conducting the appropriate procedure have been​ satisfied, what proportion of registered voters is in favor of a tax increase to reduce the federal​ debt? Explain which statistical procedure would most likely be used for the research objective given.

The correct procedure is a confidence interval for a single proportion. The goal is to determine the proportion of the population that favors a tax increase. There is no comparison being made and there is only one​ population, so rather than hypothesis​ testing, it is appropriate to use a confidence interval.

Assuming all model requirements for conducting the appropriate procedure have been​ satisfied, is the mean IQ of the students in the​ professor's statistics class higher than that of the general​ population, 100? Explain what statistical procedure should be used for this research objective.

The correct procedure is a hypothesis test for a single mean. The comparison is between the mean IQ of the class and the national average IQ. The class is a sample of the​ population, so it is not a comparison between two population means. The objective is to find whether the sample mean is higher than the population​ mean, so it is a hypothesis test and not a confidence interval.

Determine whether the following sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A researcher wishes to compare academic aptitudes of pharmacists and non-pharmacists. She obtains a random sample of 288 professionals of each category who take an academic aptitude test and determines each individual's academic aptitude.

The sampling is independent because an individual selected for one sample does not dictate which individual is to be in the second sample. The variable is qualitative because it classifies the individual.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Six years​ ago, 11.9​% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.

There is not sufficient evidence to conclude that the percentage of teenage mothers has increased.

State the conclusion based on the results of the test. According to the​ report, the standard deviation of monthly cell phone bills was ​$48.58 three years ago. A researcher suspects that the standard deviation of monthly cell phone bills is higher today. The null hypothesis is not rejected.

There is not sufficient evidence to conclude that the standard deviation of monthly cell phone bills is higher than its level three years ago of $48.58.

The head of institutional research at a university believed that the mean age of​ full-time students was declining. In​ 1995, the mean age of a​ full-time student was known to be 27.4 years. After looking at the enrollment records of all 4934​ full-time students in the current​ semester, he found that the mean age was 27.1​ years, with a standard deviation of 7.3 years. He conducted a hypothesis of H₀​: μ=27.4 years versus H₁​: μ<27.4 years and obtained a​ P-value of 0.0020. He concluded that the mean age of​ full-time students did decline. Is there anything wrong with his​ research?

Yes, the head of institutional research has access to the entire​ population, inference is unnecessary. He can say with​ 100% confidence that the mean age has decreased. There is no need for inferential claims when the population statistics have been determined. The average age of students in the year the study is being conducted is less than the average age of students in 1995.