MATH - section 8

A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples for the two types of pets are shown below. At α=0.10​, can you reject the pet​ association's claim? Assume the population variances are equal. Assume that the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) through​ (e).

"The mean annual costs of food for dogs and cats are​ equal." =, not equal, null -1.69, 1.69 2.19 reject, is there is, reject, the same as

A sleep disorder specialist wants to test the effectiveness of a new drug that is reported to increase the number of hours of sleep patients get during the night. To do​ so, the specialist randomly selects 16 patients and records the number of hours of sleep each gets with and without the new drug. The accompanying table shows the results of the​ two-night study. Construct a​ 90% confidence interval for μd​, using the inequality **** Assume the populations are normally distributed. Calculate d for each patient by subtracting the number of hours of sleep with the drug from the number without the drug. The confidence interval is

-1.96 and -1.35

Use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance α​, and sample sizes n1 and n2. Assume that the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) and​ (b). Ha​: μ1<μ2​, α=0.025​, n1=13​, n2=11

-2.074 -2.228

Use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance α​, and sample sizes n1 and n2. Assume that the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) and​ (b). Ha​:μ1≠μ2​, α=0.01​, n1=12​, n2=10

-2.845, 2.845 -3.250, 3.250

Use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance α​, and sample sizes n1 and n2. Assume that the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) and​ (b). Ha​: μ1>μ2​, α=0.05​, n1=17​, n2=8

1.714 1.895

Construct a 90% confidence interval for μ1−μ2 with the sample statistics for mean cholesterol content of a hamburger from two fast food chains and confidence interval construction formula below. Assume the populations are approximately normal with unequal variances. Stats x1=150 mg, s1=3.71 mg, n1=10 x2=132 mg, s2=2.05 mg, n2=16 Confidence interval when variances are not equal x1−x2−tcs21n1+s22n2<μ1−μ2<x1−x2+tcs21n1+s22n2 d.f. is the smaller of n1−1 or n2−1

16 and 20

What conditions are necessary in order to use the dependent samples​ t-test for the mean of the differences for a population of paired​ data?

Each sample must be randomly selected from a normal population and each member of the first sample must be paired with a member of the second sample.

What conditions are necessary in order to use the​ z-test to test the difference between two population​ means?

Each population has a normal distribution with a known standard deviation. Your answer is correct. The samples must be independent. The samples must be randomly selected.

What conditions are necessary in order to use the z​-test to test the difference between two population​ proportions?

Each sample must be randomly​ selected, independent, and n1p1, n1q1, n2p2, and n2q2 must be at least five.

The results of a survey of 200 randomly selected U.S. men and 300 randomly selected U.S. women are shown in the​ figure, which displays the percentages engaged in working or socializing and communicating on an average day. At α=0.05​, can you reject the claim that the proportion of women who work is the same as the proportion of women who socialize on an average​ day?

H0​: p1=p2 Ha​: p1≠p2 z= 0 p= 1 no, insufficient, reject

Test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and​ independent, and the populations are normally distributed. ​Claim: μ1=μ2​; α=0.10. Assume σ 2/1 = σ 2/2 Sample​ statistics: x1=35.1​, s1=3.5​, n1=12 and x2=36.8​, s2=2.4​, n2= 15

H0​: μ1=μ2 Ha​: μ1≠u2 -1.50 0.147 fail to reject, is not

A researcher claims that the stomachs of blue crabs from Location A contain more fish than the stomachs of blue crabs from Location B. The stomach contents of a sample of 16 blue crabs from Location A contain a mean of 199 milligrams of fish and a standard deviation of 38 milligrams. The stomach contents of a sample of 8 blue crabs from Location B contain a mean of 189 milligrams of fish and a standard deviation of 41 milligrams. At α=0.05​, can you support the​ researcher's claim? Assume the population variances are equal. Complete parts​ (a) through​ (d) below.

H0​: μ1−μ2≤0 Ha​: μ1−μ2>0 t=0.592 P=0.2798 fail to reject, is not

Is the difference between the mean annual salaries of statisticians in Region 1 and Region 2 more than $7000? To​ decide, you select a random sample of statisticians from each region. The results of each survey are shown to the right. At α=0.10, what should you​ conclude?

H0​: μ1−μ2≤7000 Ha​: μ1−μ2>7000 Z0= 1.28 Z>1.28 Z=0.76 Fail to reject H0. At the​10% significance​level, there is insufficient evidence to support the claim that the difference between the mean annual salaries is more than $7000.

Test the claim below about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and​ dependent, and the populations are normally distributed. ​Claim: μd=​0; α=0.01. Sample​ statistics: d=3.4​, sd=8.92​, n=6

H0​: μd=0 Ha​: μd≠0 t=.93 t0= -4.03, 4.03 (use t chart) not in, fail to reject, is not

A food manufacturer claims that eating its new cereal as part of a daily diet lowers total blood cholesterol levels. The table shows the total blood cholesterol levels​ (in milligrams per deciliter of​ blood) of seven patients before eating the cereal and after one year of eating the cereal as part of their diets. Use technology to test the mean difference. Assume the samples are random and​ dependent, and the population is normally distributed. At α=0.05​, can you conclude that the new cereal lowers total blood cholesterol​ levels?

H0​: μd≤0 HA​: μd>0 t=1.433 p=.1009 fail to reject, is not or reject, is *****

Test the claim below about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and​ dependent, and the populations are normally distributed. ​Claim: μd<​0; α=0.01. Sample​ statistics: d=1.6​, sd=3.7​, n=16

H0​: μd≥0 Ha​: μd<0 t=1.73 HOW TO SOLVE FOR T: (1.3)/(3.1/squared 17)=1.73 t0=-2.60 Since the test statistic is not in the rejection​ region, fail to reject the null hypothesis. There is not statistically significant evidence to support the claim.

The mean exam score for 46 male high school students is 24.8 and the population standard deviation is 4.6. The mean exam score for 57 female high school students is 21.8 and the population standard deviation is 4.3. At α=0.01​, can you reject the claim that male and female high school students have equal exam​ scores? Complete parts​ (a) through (e).

Male and female high school students have equal exam scores. H0​: μ1=μ2 Ha​: μ1≠u2 -2.58, 2.58 z<−​2.58, z>2.58 z=3.39 Reject H0. The standardized test statistic is in the rejection region. 1, sufficient, reject, equal to

A scientist claims that pneumonia causes weight loss in mice. The table shows the weights​ (in grams) of six mice before infection and two days after infection. At α=0.10​, is there enough evidence to support the​ scientist's claim? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (e) below.

Pneumonia causes weight loss in mice. ​HO: μd≤0 Ha​: μd>0 T> d=.9 sd=0.358 t=6.16 reject, is, support, pneumonia, weight loss

Use the technology display to make a decision to reject or fail to reject the null hypothesis at α=0.01. Make the decision using the standardized test statistic and using the​ P-value. Assume the sample sizes are equal.

Reject the null hypothesis because the​ P-value is less than the level of significance.

Use the technology display to make a decision to reject or fail to reject the null hypothesis at α=0.05. Make the decision using the standardized test statistic and using the​ P-value. Assume the sample sizes are equal.

Reject the null hypothesis because the​P-value is less than the level of significance.

Explain how to perform a​ two-sample z-test for the difference between two population means using independent samples with σ1 and σ2 known.

State the hypotheses and identify the claim. Specify the level of significance. Find the critical​ value(s) and identify the rejection​ region(s). Find the standardized test statistic. Make a decision and interpret it in the context of the claim.

A real estate agency says that the mean home sales price in​ Casper, Wyoming is the same as in​ Cheyenne, Wyoming. The mean home sales price for 35 homes in​ Casper, Wyoming is ​$349,237. Assume the population standard deviation is $169,306. The mean home sales price for 41 homes in​ Cheyenne, Wyoming is ​$435,244. Assume the population standard deviation is $162,540. At α=0.05​, is there enough evidence to reject the​ agency's claim? Complete parts​ (a) through​ (e).

The mean home sales price in​ Casper, Wyoming is the same as in​ Cheyenne, Wyoming. H0​: μ1 = μ2 Ha​: μ1 ≠ u2 The critical values are z0=±1.96 Z<-1.96 or Z> 1.96 Z=-2.25 P=0.025 reject, less than or equal to is, reject, the same as

A marine biologist claims that the mean length of mature female pink seaperch is different in fall and winter. A sample of 10 mature female pink seaperch collected in fall has a mean length of 114 millimeters and a standard deviation of 9 millimeters. A sample of 2 mature female pink seaperch collected in winter has a mean length of 101 millimeters and a standard deviation of 12 millimeters. At α=0.20​, can you support the marine​ biologist's claim? Assume the population variances are equal. Assume the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) through​ (e) below.

The mean length of mature female pink seaperch is different in fall and​ winter." The null​ hypothesis, H0​, is mu 1 equals mu 2 .μ1=μ2.The alternative​ hypothesis, Ha​, is mu 1 not equals mu 2 . The alternative​ hypothesis,Ha 1.372, -1.372 t<-t0, t>t0 t=1.796 reject there is

A group of third grade students is taught using a new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the​ back-to-back stem-and-leaf plot. At α=0.05​, is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method​ does? Assume the population variances are equal. Complete parts​ (a) through​ (e) below. Assume the samples are random and​ independent, and the populations are normally distributed.

The new method of teaching reading produces higher reading test scores than the old​ method." The null​ hypothesis, H0​, is mu 1 greater than or equals mu 2 .μ1≥μ2. The alternative​ hypothesis, Ha​, is mu 1 less than mu 2 .μ1<μ2. The alternative​ hypothesis, Ha -1.682 t<-t0 t=-4.037 reject there is

The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020-2021 season are shown in the table. At α=0.05​, is there enough evidence to support the claim that passing play percentage is different for home and away​ games? Assume the samples are random and​ dependent, and the populations are normally distributed. Complete parts​ (a) through​ (f).

The passing play percentages have changed. H0​: μd=0 Ha​: μd≠0 d=-4 sd=7.081 t=-1.79 p=.108 fail to reject, yes, are is not, support, changed

What conditions are necessary in order to use the​ t-test for testing the difference between two population​ means?

The population standard deviations are unknown. The samples are randomly selected and independent. The populations are normally distributed or each sample size is at least 30.

Determine whether a normal sampling distribution can be used for the following sample statistics. If it can be​ used, test the claim about the difference between two population proportions p1 and p2 at the level of significance α. Assume that the samples are random and independent. ​Claim: p1≠p2​, α=0.01 Sample​ Statistics: x1=40​, n1=69​, x2=38​, n2=59

The samples are random and independent. A normal sampling distribution can be used because n1p=42.0542.05​ n1q=26.9526.95​ n2p=35.9535.95​, n2q=23.0523.05. H0​: p1=p2 Ha​: p1≠p2 Z=-.74 P=.457 Since P>α​, fail to reject H0. There is not enough evidence at the α=0.01 level of significance to support the claim.

Classify the two given samples as independent or dependent. Sample​ 1: The average time for 40 amateur skiers to finish a race Sample​ 2: The average time for the same 40 amateur skiers to finish a race after

The two given samples are dependent because the same skiers were sampled.

Explain why the null hypothesis H0: μ1=μ2 is equivalent to the null hypothesis H0: μ1−μ2=0.

They are equivalent through algebraic manipulation.

What is the difference between two samples that are dependent and two samples that are​ independent? Give an example of each.

Two samples are dependent when: each member of one sample corresponds to a member of the other sample. One example is: the weights of 22 people before starting an exercise program and the weights of the same 22 people 6 weeks after starting the exercise program. Two samples are independent when: the sample selected from one population is not related to the sample selected from the other population. One example is: the weights of 25 cats and the weights of 20 dogs.

physical therapist claims that one​ 600-milligram dose of Vitamin C will increase muscular endurance. The table available below shows the numbers of repetitions 15 males made on a hand dynamometer​ (measures grip​ strength) until the grip strengths in three consecutive trials were​ 50% of their maximum grip strength. At α=0.05​, is there enough evidence to support the ​ therapist's claim? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (e) below.

Using Vitamin C increases muscular endurance. ​H0: μd≥0 Ha​: μd<0 t<-1.761 d= -197.9 sd=142.9 t=-5.364 reject, is , support, vit c, increases

A company wants to determine whether its consumer product ratings ​(0−​10) have changed from last year to this year. The table below shows the​ company's product ratings from eight consumers for last year and this year. At α=​0.05, is there enough evidence to conclude that the ratings have​ changed? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (f).

changed H0​: μd=0 Ha​: μd≠0 t0= -2.365, 2.365 t<-2.365, t>2.365 d=1.25 sd=1.982 t=1.784 fail to reject the null hypothesis At the​ 5% significance​ level, there is not enough evidence to support the claim that the product ratings have changed from last year to this year.

Use the figure to the​ right, which shows the percentages of adults from several countries who favor building new nuclear power plants in their country. The survey included random samples of 1076 adults from Country​ A, 1120 adults from Country​ B, 1018 adults from Country​ C, and 1016 adults from Country D. At α=0.07​, can you support the claim that the proportion of adults in Country B who favor building new nuclear power plants in their country is different from the proportion of adults from Country C who favor building new nuclear power plants in their​ country? Assume the random samples are independent. A bar graph has a vertical axis labeled from 0 to 100 in increments of 10. There are vertical bars with labels and heights in percentages as follows: Country A, 51; Country B, 49; Country C, 46; Country D, 35.020406080100Country A 51%Country B 49%Country C 46%Country D 35%

different than H0​: p1=p2 Ha​: p1≠p2 z=1.39 P= .165 Fail to reject H0 because the​P-value is greater than the significance level α. No​, at the 7​% significance​level, there is insufficient evidence to support the claim.

In a survey of 490 drivers from the​ South, 395 wear a seat belt. In a survey of 350 drivers from the​ Northeast, 282 wear a seat belt. At α=0.07​, can you support the claim that the proportion of drivers who wear seat belts is greater in the South than in the​ Northeast? Assume the random samples are independent. Complete parts​ (a) through​ (e).

greater than ​Ho: p1≤p2 Ha​: p1>p2 1.48 z>1.48 .01 Fail to reject H0 because the test statistic is not in the rejection region. At the 7​% significance​ level, there is insufficient evidence to support the claim.

Decide whether the normal sampling distribution can be used. If it can be​ used, test the claim about the difference between two population proportions p1 and p2 at the given level of significance α using the given sample statistics. Assume the sample statistics are from independent random samples. ​Claim: p1<p2​, α=0.01 Sample​ statistics: x1=35​, n1=70 and x2=38​, n2=59

is, is, is, is, can H0​: p1=p2 Ha​: p1<p2 -Z0= -2.33 Z= -1.64 Fail to reject H0. There is insufficient evidence that there is a difference between p1 and p2.

Use the figure to the​ right, which shows the percentages of adults from several countries who favor building new nuclear power plants in their country. The survey included random samples of 1015 adults from Country​ A, 1052 adults from Country​ B, 1110 adults from Country​ C, and 1022 adults from Country D. At α=0.07​, can you reject the claim that the proportion of adults in Country A who favor building new nuclear power plants in their country is the same as the proportion of adults from Country B who favor building new nuclear power plants in their​ country? Assume the random samples are independent.

same as H0​: p1=p2 Ha​: p1≠p2 z=-.45 P= .649 Fail to reject H0 because the​ P-value is greater than the significance level α. No​, at the 7​% significance​level, there is insufficient evidence to reject the claim.

In a survey of 185 females who recently completed high​ school, 80​% were enrolled in college. In a survey of 175 males who recently completed high​ school, 68​% were enrolled in college. At α=0.09​, can you reject the claim that there is no difference in the proportion of college enrollees between the two​ groups? Assume the random samples are independent. Complete parts​ (a) through​ (e).

the same as H0​: p1=p2 Ha​: p1≠p2 -1.70, 1.70 z<-1.70 and z> 1.70 z=2.6 Reject H0 because the test statistic is in the rejection region. At the 9​% significance​ level, there is sufficient evidence to reject the claim.

Decide whether the normal sampling distribution can be used. If it can be​ used, test the claim about the difference between two population proportions p1 and p2 at the given level of significance α using the given sample statistics. Assume the sample statistics are from independent random samples. ​Claim: p1=p2​, α=0.05 Sample​ statistics: x1=65​, n1=110 and x2=167​, n2=215

yes H0​: p1=p2 Ha​: p1≠p2 -1.96, 1.96 -Z0= -1.96 and Z0 = 1.96 z= -3.51 Reject H0. There is sufficient evidence that there is a difference between p1 and p2.

A personnel director in a particular state claims that the mean annual income is the same in one of the​ state's counties​ (County A) as it is in another county​ (County B). In County​ A, a random sample of 17 residents has a mean annual income of $42,500 and a standard deviation of $8900. In County​ B, a random sample of 14 residents has a mean annual income of $38,300 and a standard deviation of $5500. At α=0.05​, answer parts​ (a) through​ (e). Assume the population variances are not equal. Assume the samples are random and​ independent, and the populations are normally distributed.

​"The mean annual incomes in Counties A and B are​ equal." u1=u2 u1 doesnt = u2 the null hypothesis, h0 2.160, -2.160 t,<t0, t>t0 t=1.608 fail to reject there is not

To compare the dry braking distances from 30 to 0 miles per hour for two makes of​ automobiles, a safety engineer conducts braking tests for 35 models of Make A and 35 models of Make B. The mean braking distance for Make A is 42 feet. Assume the population standard deviation is 4.8 feet. The mean braking distance for Make B is 45 feet. Assume the population standard deviation is 4.5 feet. At α=0.10​, can the engineer support the claim that the mean braking distances are different for the two makes of​ automobiles? Assume the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) through​ (e).

​(a) Identify the claim and state H0 and Ha. -The mean braking distance is different for the two makes of automobiles. What are H0 and Ha​? H0​: μ1=μ2 Ha​: μ1≠ u2 critical value: -1.64, 1.64 rejection region: z<−1.64​, z> 1.64 Z=-2.70 Reject H0. The standardized test statistic falls in the rejection region. 10, sufficient, support, different from

Is the difference between the mean annual salaries of entry level architects in​ Denver, Colorado, and​ Lincoln, Nebraska, equal to ​$9000​? To​ decide, you select a random sample of entry level architects from each city. The results of each survey are shown. Assume the population standard deviations are σ1=​$6516 and σ2=​$6255. At α=0.01​, what should you​ conclude?

​H0: μ1−μ2=9000 Ha​: μ1−μ2≠9000 Z= -3.19 P=0 Reject H0. At the​ 1% level of​ significance, there is sufficient evidence to reject the claim that the difference in mean annual salaries is ​$9000.

Test the claim below about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and​ dependent, and the populations are normally distributed. ​Claim: μd≥​0; α=0.01. Sample​ statistics: d=−1.9​, sd=1.1​, n=19

​HO: μd≥0 Ha​: μd<0 t=-7.53 p=0 less than or equal to, reject, is