AP Stats Chapter 9

Type 1 error

You reject the null hyp when it is actually true.

The P-value of a test of a null hypothesis is the probability that\ a) assuming the null hypothesis is true, the test statistic will take a value at least as extreme as that observed. b) assuming the null hypothesis is false, the test statistic will take a value at least as extreme as that actually observed. c) the null hypothesis is true. d) the null hypothesis is false.

a) assuming the null hypothesis is true, the test statistic will take a value at least as extreme as that observed.

An opinion poll asks a simple random sample of 100 college seniors how they view their job prospects. In all, 53 say "good." Does the poll give convincing evidence to conclude that more than half of all seniors think their job prospects are good? If p = the proportion of all college seniors who say their job prospects are good, what are the hypotheses for a test to answer this question? a. Ho: p = 0.5, Ha: p > 0.5. b. Ho: p > 0.5, Ha: p = 0.5 c. Ho: p = 0.5, Ha: p ≠ 0.5. d. Ho: p = 0.5, Ha: p < 0.5. e. Ho: p ≠ 0.5, Ha: p > 0.5.

a. Ho: p = 0.5, Ha: p > 0.5.

An appropriate 95% confidence interval for μ has been calculated as (0.73, 1.92 ) based on n = 15 observations from a population with a Normal distribution. If we wish to use this confidence interval to test the hypothesis Ho: μ = 0 against Ha: μ ≠ 0, which of the following is a legitimate conclusion? a. Reject Ho at the α = 0.05 level of significance. b. Fail to reject Ho at the α = 0.05 level of significance. c. Reject Ho at the α = 0.10 level of significance. d. Fail to reject Ho at the α = 0.10 level of significance. e. We cannot perform the required test since we do not know the value of the test statistic.

a. Reject Ho at the α = 0.05 level of significance.

One response variable was "volume of nasal secretions" (if you have a cold, you blow your nose a lot). Take the average volume of nasal secretions in people without colds to be μ = 1. An increase to μ = 3 indicates a cold. Which of the following describes the significance level of a test of Ho :μ = 1 versus Ha :μ > 1? a. The probability that the test rejects Ho when μ = 1 is true. b. The probability that the test rejects Ho when μ = 3 is true. c. The probability that the test fails to reject Ho when μ = 3 is true. d. The probability that the test fails to reject Ho when μ = 1 is true. e. None of the above

a. The probability that the test rejects Ho when μ = 1 is true.

If a significance test gives a P-value of 0.005, a. the null hypothesis is very likely to be true. b. we do not have convincing evidence in favor of the null hypothesis. c. we do not have convincing evidence against the null hypothesis. d. we do have convincing evidence against the null hypothesis.

d. we do have convincing evidence against the null hypothesis.

We want to test Ho: µ = 1.5 vs. Ha : µ ≠ 1.5 at α = 0.05. A 95% confidence interval for µ calculated from a given random sample is (1.4, 3.6). Based on this finding we (a) fail to reject Ho . (b) reject Ho . (c) cannot make any decision at all because the value of the test statistic is not available. (d) cannot make any decision at all because the distribution of the population is unknown. (e) cannot make any decision at all because (1.4, 3.6) is only a 95% confidence interval for µ.

(a) fail to reject Ho .

The recommended daily Calcium intake for women over 21 (and under 50) is 1000 mg per day. The health services at a college are concerned that women at the college get less Calcium than that, so they take a random sample of female students in order to test the hypotheses Ho: µ =1000 versus Ha: µ < 1000. Prior to the study they estimate that the power of their test against the alternative Ha: µ= 900 is 0.85. Which of the following is the best interpretation of this value? (a) The probability of making a Type II error. (b) The probability of rejecting the null hypothesis when the parameter value is 1000. (c) The probability of rejecting the null hypothesis when the parameter value is 900. (d) The probability of failing to reject the null hypothesis when the parameter value is 1000. (e) The probability of failing to reject the null hypothesis when the parameter value is 900.

(c) The probability of rejecting the null hypothesis when the parameter value is 900.

For a Proportion P H A N T O M

(p̂-Ho) = z* √( Ho(1-Ho)/n )

Lumber companies dry freshly-cut wood in kilns before selling it. As a result of the drying process a certain percentage of the boards become "checked," which means that cracks develop at the ends of the boards. The current drying procedure for 1" x 4" red oak boards is known to produce cracks in 16% of the boards. The drying supervisor at a lumber company wants to test a new method to determine if fewer boards crack. Suppose the drying supervisor uses the new method on an SRS of boards and finds that the sample proportion of checked boards is 0.11, which produces a P-value of 0.027. (b) What conclusion would you draw at the a = 0.05 level?

0.027<0.05 we have sufficent evdience that we can reject the null hyp. ergo we can support the claim that the true proportion of all check board procude at the factory is less than 0.16

Lumber companies dry freshly-cut wood in kilns before selling it. As a result of the drying process a certain percentage of the boards become "checked," which means that cracks develop at the ends of the boards. The current drying procedure for 1" x 4" red oak boards is known to produce cracks in 16% of the boards. The drying supervisor at a lumber company wants to test a new method to determine if fewer boards crack. Suppose the drying supervisor uses the new method on an SRS of boards and finds that the sample proportion of checked boards is 0.11, which produces a P-value of 0.027. (c) What conclusion would you draw at the a = 0.01 level?

0.027>0.01 we have insuffiencent evidene to reject the null ergo we can can't support the claim that the true proportion of all check board procude at the factory is less than 0.16

What is Null Hypothesis

Ho Assuming nothing changes (always an equal sign)

For a Proportion P H A N T O M

Ho: p= Ha: p(<,>,≠)

For a mean P H A N T O M

Ho: μ= Ha: μ(<,>,≠)

To increase the power of the test (higher the better)

Increase the sample size Increase the level of significance Increase the difference between the null and alternate hypothesis parameter values

How can we decide if null is wrong or it was random chance

Increase the sample size and or the # of samples significance testing

Based on B and C would alpha 0.1 or 0.01 be a better siginificance level for this text

Increasing alpha increases the power of the test and we would have less chance cancelling a show people actually watch it

Is the ratio of male births to female births even? A simple random sample of births in a major metropolitan area found 1345 boys among 2546 firstborn children. A 99% confidence interval for =the proportion of male births in this population is given by (0.5028, 0.5538). (b) What information is provided by the confidence interval that would not be provided by a test of significance alone?

It give us a interval at possible proportion of males births in this population

What is alternative hypothesis

It uses equalities (<,>, not equal to)

For a Proportion P H A N T O M

One proportion z test

For a Proportion P H A N T O M

P( z(<,>)z* ) normalcdf(LB,UB,0,1)= p* (if two tailed p*x2) p*>0.05 we fail to reject null p*<0.05 we reject null

For a mean P H A N T O M

P(t<,>t*) tcdf(LB,UB,df)= μ* (if two tailed μ*x2) μ*>0.05 we fail to reject null μ*<0.05 we reject null

LeRoy, a starting player for a major college basketball team, made only 40% of his free throws last season. During the summer, he worked on developing a softer shot in hopes of changing his free throw accuracy. In the first eight games of this season, LeRoy made 25 free throws in 40 attempts. You want to investigate whether LeRoy's work over the summer will result in a different proportion of free-throw successes this season. What conclusion would you draw at the α = 0.01 level about LeRoy's free throw shooting? Justify your answer with a complete significance test.

P: p= the true proportion that all successful free throws made by LeRoy this season H: Ho: p=0.4 Ha: p≠0.4 p̂(25/40)=0.625 A: No mention of randomness so we are going to assume and continue✓ 10% rule✓ np>10✓ n(1-p)>10✓ N: one proportion z-test T: (0.625-0.4) = 2.90 √( .04(1-.04)/40 ) O: P( z>2.90 ) normalcdf(2.90,1000000,0,1)= 0.00187 x2 =0.003676 0.003676<0.01 we reject null M: -We have sufficient evidence to reject the null ergo we can support the claim the true proportion that all successful free throws made by LeRoy this season is greater than 0.4 is true, based on our sample at a level of significance of α=0.01.

Economists often track employment trends by measuring the proportion of people who are "underemployed," meaning they are either unemployed or would like to work full time but are only working part-time. In the summer of 2013, 17.6% of Americans were "underemployed." The mayor of Thicksburg wants to show the voters that the situation is not as bad in his town as it is in the rest of the country. His staff takes a simple random sample of 300 Thicksburg residents and finds that 45 of them are underemployed. (a) Do the data give convincing evidence that the proportion of underemployed in Thicksburg is lower than elsewhere in the country? Support your answer with a significance test.

P: p= true proportion of underemployed residents in Thinksburg H: Ho: p=0.176 Ha: p<0.176 p̂=(45/300)=0.15 A: np<10✓ n(1-p)<10✓ 10% rule✓ Random sample✓ N: 1 proportion z test T: (0.15-0.176) = -1.183 √( 0.176(1-0.176)/300 ) O: P( z <-1.183 ) normalcdf(-100000,-1.183,0,1)= 0.1185 0.1185>0.05 we fail to reject the null M: -We have insufficient evidence to reject the null ergo we cannot support the claim that the true proportion of underemployed residents in Thinksburg less than the national avg of 0.176 is true, based on our sample at level of significance of α=0.05.

Sweet corn of a certain variety is known to produce individual ears of corn with a mean weight of 8 ounces. A farmer is testing a new fertilizer designed to produce larger ears of corn, as measured by their weight. He finds that 32 randomly-selected ears of corn grown with this fertilizer have a mean weight of 8.23 ounces and a standard deviation of 0.8 ounces. There are no outliers in the data. Do these samples provide convincing evidence at the a= 0.05 level that the fertilizer had a positive impact on the weight of the corn ears? Justify your answer.

P: μ= the true mean weight of all ears of sweet corn grown at this farm H: Ho: μ=8 Ha: μ > 8 A: Random sample ✓ 10% rule ✓ 30≤32 by CLT ✓ N: one sample t-test T: 8.23-8 =1.626 (0.8/√32) O: P(t<1.626) tcdf(1.626,100000000,31)= 0.057 0.057>0.05 we fail to reject the null M: -We have insufficient evidence to reject the null ergo we cannot support the claim the true mean weight of all ears of sweet corn grown at this farm is greater than 8 is true, based on our sample at a level of significance of α=0.05.

Power of a test is the

Probability we reject Ho, when Ha is true = 1-β (This is what we want)

For a mean P H A N T O M

Random sample ✓ 10% rule ✓ n ≥ 30 or We are told the population distribution is normal or We create a normal probability plot and show the population is normal with no outliers and the probability plot is linear

For a Proportion P H A N T O M

Random sample ✓ (if no mention state that and that we assume it is random) 10% rule ✓ np>10 n(1-p)>10 ✓

For a Proportion P H A N T O M

Reject null -We have sufficient evidence to reject the null ergo we can support the claim HA is true, based on our sample at level of significance of α=#. Fail to reject null -We have insufficient evidence to reject the null ergo we cannot support the claim HA is true, based on our sample at level of significance of α=#.

For a mean P H A N T O M

Reject null -We have sufficient evidence to reject the null ergo we can support the claim Ha is true, based on our sample at level of significance of α=#. Fail to reject null -We have insufficient evidence to reject the null ergo we cannot support the claim HA is true, based on our sample at level of significance of α=#.

You test the hypothesis against the alternative and obtain a P-value of 0.022. Which of the following must be true? a. A 95% confidence interval for μ will include the value 1. b. A 95% confidence interval for μ will include the value 0. c. A 99% confidence interval for μ will include the value 1. d. A 99% confidence interval for μ will include the value 0.

c. A 99% confidence interval for μ will include the value 1.

one sided alternative hypothesis is determined by

Shows that the real parameter is greater or less than the assumed parameter.

two sided alternative hypothesis is determined by

Shows that the real parameter is not equal to the assumed parameter.

When the manufacturing process is working properly, Never Ready batteries have lifetimes that follow a slightly right-skewed distribution with μ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded. (b) Since testing the lifetime of a battery requires draining the battery completely, the supervisor wants to sample as few batteries as possible from each hour's production. She is considering a sample size of n = 4. Explain why this sample size may lead to problems in carrying out the significance test from A.

Since the population is slightly skewed right a sample of 4 is not enough for the sample to be from a population that is approximately normal

What is P value

The probability that getting the sample data that we got given that the null hypothesis is true

interperate P-Value template

There is (p-value)% chance that the true proportion of (context) is really (first percentage) of getting a proportion of (send percentage)

You construct a 95% confidence interval for a mean and find it to be 1.1 ± 0.8. Which of the following is true? a. A test of the hypotheses Ho: μ = 1.2, Ha: μ ≠ 1.2 would reject Ho at the 0.05 level. b. A test of the hypotheses Ho: μ = 1.1, Ha: μ ≠ 1.1 would reject Ho at the 0.05 level. c. A test of the hypotheses Ho: μ = 0, Ha: μ ≠ 0 would reject Ho at the 0.05 level. d. All three tests above would reject Ho at the 0.05 level.

c. A test of the hypotheses Ho: μ = 0, Ha: μ ≠ 0 would reject Ho at the 0.05 level.

Economists often track employment trends by measuring the proportion of people who are "underemployed," meaning they are either unemployed or would like to work full time but are only working part-time. In the summer of 2013, 17.6% of Americans were "underemployed." The mayor of Thicksburg wants to show the voters that the situation is not as bad in his town as it is in the rest of the country. His staff takes a simple random sample of 300 Thicksburg residents and finds that 45 of them are underemployed. (b) Interpret the P-value from your test in the context of the problem.

There is a 11.85% chance we would get a sample of 15% of underemployed Thicksburg residents if the true proportion of underemployed residents is actually 17.6%

Lumber companies dry freshly-cut wood in kilns before selling it. As a result of the drying process a certain percentage of the boards become "checked," which means that cracks develop at the ends of the boards. The current drying procedure for 1" x 4" red oak boards is known to produce cracks in 16% of the boards. The drying supervisor at a lumber company wants to test a new method to determine if fewer boards crack. Suppose the drying supervisor uses the new method on an SRS of boards and finds that the sample proportion of checked boards is 0.11, which produces a P-value of 0.027. (a) Interpret the P-value in the context of the problem.

There is a 2.7% chance that the true proportion of checked boards at the factory is really 16% of getting a proportion of 11%

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test Ho: μ=5 Ha: μ≠5 If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level against the alternative μ = 5.1 is 0.23. (a) Explain in simple language what "power = 0.23" means in this setting.

There is a 23% chance that we conclude μ≠5 when really μ=5.1

The network detemines that the sample size of 2000 the power of this test at a 5% signifcance level for Ha: p=0.14 is only 0.39. explain what the power of the test measures in the contex of the problem

There is a 39% chance of concluding more than 12% of people watch the show when 14% of people actually watch the show

When the manufacturing process is working properly, Never Ready batteries have lifetimes that follow a slightly right-skewed distribution with μ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded. (d) The quality control officer is considering changing the significance level of the test to 1%. Discuss the impact this might have on error probabilities and the power of the test, and describe the practical consequences of this change.

This would decrease the probability of a type 1 error but increase the probability of a type 2 error. This would increase the chance of selling bad batteries but decrease the chance of throwing batteries away

When the manufacturing process is working properly, Never Ready batteries have lifetimes that follow a slightly right-skewed distribution with μ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded. (c) Describe a Type I and a Type II error in this situation and the consequences of each.

Type 1: we conclude the mean lifetime of NeverReady batteries is less than 7hrs when actually it is 7 hours Consequence- The supervisor will through out good batteries by mistake Type 2: We conclude the mean lifetime of NeverReady batteries is 7hrs when actually it is less than 7hrs Consequence- The company will sell batteries that claim 7hrs and customers will complain

14.A certain cigarette brand advertises that the mean nicotine content of their cigarettes is 1.5 mg, but you are suspicious and plan to investigate the advertised claim by testing the hypotheses Ho:μ = 1.5 versus Ha:μ > 1.5 at the α = 0.05 significance level. You will do so by measuring the nicotine content of 30 randomly selected cigarettes of this brand. (c) From the perspective of public health, which error—Type I or Type II—is more serious? Explain.

Type 2 because failing to reject the level being 1.5mg when it is greater than 1.5mg, which is a false advertisement and can cause a lot more diseases and damage and increase the severity of the diseases and damage done

14.A certain cigarette brand advertises that the mean nicotine content of their cigarettes is 1.5 mg, but you are suspicious and plan to investigate the advertised claim by testing the hypotheses Ho:μ = 1.5 versus Ha:μ > 1.5 at the α = 0.05 significance level. You will do so by measuring the nicotine content of 30 randomly selected cigarettes of this brand. (b) Describe what a Type II error would be in this context.

We fail to reject level being 1.5mg when it is really greater than 1.5mg, which is a false advertisement. We can see an increase in cigarette use and an increase in disesese cause by citgarette

Type 2 error

You fail to reject the null hyp when Ha is actually true

Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually vary slightly from bag to bag and are Normally distributed with mean μ. A representative of a consumer advocacy group wishes to see if there is any evidence that the mean net weight is less than advertised and so intends to test the hypotheses Ho: μ=14 Ha: μ<14 a. Type I error in this situation would mean a. concluding that the bags are being underfilled when they actually aren't. b. concluding that the bags are being underfilled when they actually are. c. concluding that the bags are not being underfilled when they actually are. d. concluding that the bags are not being underfilled when they actually aren't. e. none of these

a. Type I error in this situation would mean a. concluding that the bags are being underfilled when they actually aren't.

In formulating hypotheses for a statistical test of significance, the null hypothesis is often a. a statement of "no effect" or "no difference." b. the probability of observing the data you actually obtained. c. a statement that the data are all 0. d. 0.05.

a. a statement of "no effect" or "no difference."

If we reject the null hypothesis when, in fact, it is true, we have a. committed a Type I error. b. committed a Type II error. c. a probability of being correct that is equal to the P-value. d. a probability of being correct that is equal to 1 - P-value.

a. committed a Type I error.

What is a significance level?

alpha= the cuff off % to say that something is statistically too unlikly to happen by chance

Which of the following statements is/are correct? I. The power of a significance test depends on the effect size. II. The probability of a Type II error is equal to the significance level of the test. III. Error probabilities can be expressed only when a significance level has been specified. a. I and II only b. I and III only c. II and III only d. I, II, and III e. None of the above

b. I and III only

Which of the following is not a required condition for performing a t-test about an unknown population mean μ? a. The data can be viewed as a simple random sample from the population of interest. b. The population standard deviation σ is known. c. The population distribution is Normal or the sample size is large (say n > 30). d. The data represent n independent observations e. All four of the above are required conditions.

b. The population standard deviation σ is known.

You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect that the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct? a. You should not use the t procedure because the population does not have a normal distribution. b. You may use the t procedure provided your sample size is at least thirty. c. You may use the t procedure, but you should probably claim only that the significance level is 0.10. d. You may not use the t procedure. t procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses.

b. You may use the t procedure provided your sample size is at least thirty.

A Type II error is a. rejecting the null hypothesis when it is true. b. failing to reject the null hypothesis when it is false. c. incorrectly specifying the null hypothesis. d. incorrectly specifying the alternative hypothesis.

b. failing to reject the null hypothesis when it is false.

In a test of Ho: p = 0.7 against Ha: p ≠ 0.7, a sample of size 80 produces z = 0.8 for the value of the test statistic. Which of the following is closest to the P-value of the test? a. 0.2090 b. 0.2119 c. 0.4238 d. 0.4681 e. 0.7881

c. 0.4238

A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean GPA (grade point average) for male and female scholarship athletes (P = 0.287). Which of the following is a correct interpretation of this value? a. The GPAs for male and female scholarship athletes are identical, except for 28.7% of the athletes. b. the maximum difference in GPAs between male and female scholarship athletes is 0.287. c. If in fact there is no difference in mean GPAs, the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample is 0.287. d. The chance that a pair of randomly chosen male and female scholarship athletes would have a significant difference in GPAs is 0.287.

c. If in fact there is no difference in mean GPAs, the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample is 0.287.

The water diet requires you to drink two cups of water every half hour from the time you get up until you go to bed, but otherwise allows you to eat whatever you like. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are What would a Type II error be for this test of the water diet? a. Concluding that the diet leads to weight loss when it doesn't. b. Concluding that the diet leads to weight loss when it really does c. Not concluding that the diet leads to weight loss when it does. d. Not concluding that the diet leads to weight loss when it really doesn't. e. Drawing a conclusion from this test when the Normality condition has not been satisfied.

c. Not concluding that the diet leads to weight loss when it does.

Some people say that more babies are born in September than in any other month. To test this claim, you take a simple random sample of 150 students at your school and find that 21 of them were born in September. You are interested in whether the proportion born in September is higher than 1/12—what you would expect if September was no different from any other month. Thus your null hypothesis is Ho :p =1/12 . The P-value for your test is 0.0056. Which of the following statements best describes what the P-value measures? a. The probability that September birthdays are no more common that any other month is 0.0056. b. The probability that September birthdays are more common is 0.0056 c. The probability that the proportion of September birthdays in the population is not equal to 1/12 is 0.0056 d. 0.0056 is the probability of getting a sample with a proportion of September birthdays this far or farther above 1/12 if the

d. 0.0056 is the probability of getting a sample with a proportion of September birthdays this far or farther above 1/12 if the true proportion is 1/12

A researcher wishes to determine if people are able to complete a certain pencil and paper maze more quickly while listening to classical music. Suppose previous research has established that the mean time needed for people to complete a certain maze (without music) is 40 seconds. The researcher, therefore, decides to test the hypotheses Ho :μ = 40 versus Ha:μ < 40, where m = the time in seconds needed to complete the maze while listening to classical music. a. The researcher has proved that listening to classical music substantially improves the time it takes to complete the maze. b. The researcher has strong evidence that listening to classical music substantially improves the time it takes to complete the maze. c. The researcher has moderate evidence that listening to classical music substantially improves the time it takes to complete the maze. d. Although the researcher has obtained a statistically significant

d. Although the researcher has obtained a statistically significant result, it appears to have little practical significance.

Which of the following increases the power of a significance test? a. Using a two-tailed test instead of a one-tailed test. b. Decreasing the size of your sample c. Finding a way to increase the population standard deviation σ . d. Increasing the significance level α. e. Decrease the effect size.

d. Increasing the significance level α.

A test of significance produces a P-value of 0.024. Which of the following conclusions is appropriate? a. Accept Ha at the α = 0.05 level b. Reject Ha at the α = 0.01 level c. Fail to reject Ho at the α = 0.05 level d. Reject Ho at the α = 0.05 level

d. Reject Ho at the α = 0.05 level

In a test of Ho: μ = 100 against Ha: μ ≠ 100, a sample of size 10 produces a sample mean of 103 and a P-value of 0.08. Which of the following is true at the 0.05 level of significance? a. There is sufficient evidence to conclude that μ ≠ 100. b. There is sufficient evidence to conclude that μ = 100. c. There is insufficient evidence to conclude that μ = 100. d. There is insufficient evidence to conclude that μ ≠ 100. e. There is sufficient evidence to conclude that μ > 103.

d. There is insufficient evidence to conclude that μ ≠ 100.

I conduct a statistical test of hypotheses and find that the null hypothesis is statistically significant at level α = 0.05. I may conclude that a. the test would also be significant at level α = 0.10. b. the test would also be significant at level α = 0.01. c. the P-value is less than .05. d. both (A) and (C) are true.

d. both (A) and (C) are true.

A significance test was performed to test the null hypothesis Ho: μ = 2 versus the alternative Ha: μ ≠ 2. A sample of size 28 produced a test statistic is t = 2.051. Assuming all conditions for inference were met, which of the following intervals contains the P-value for this test? a. P<0.01 b. 0.01≤ P <0.02 c. 0.02 ≤ P <0.025 d. 0.025 ≤ P < 0.05 e. 0.05 ≤ P < 0.10

e. 0.05 ≤ P < 0.10

A medical researcher is working on a new treatment for a certain type of cancer. After diagnosis, the average survival time on the standard treatment is two years. In an early trial, she tries the new treatment on five subjects and finds that they have an average survival time of four years after diagnosis. Although the survival time has doubled, the results of a t-test for mean survival time are not statistically significant even at the 0.10 significance level. Which of the following is the best course of action for the researcher? a. Since the test was not statistically significant, she should abandon study of this treatment and move on to more promising ones. b. She should reexamine her computations—it is likely that she made an error. c. She should increase the significance level of her test so that she rejects the null hypothesis, since the treatment clearly has a positive impact. d. She should use a z-test in

e. She should expand her research program to include more subjects—this was a very small sample.

You are testing the hypothesis that a new method for freezing green beans preserves more vitamin C in the beans than the conventional freezing method. Beans frozen by the conventional methods are known to have a mean Vitamin C level of 12 mg per serving, so you are testing Ho :μ = 12 versus Ha :μ > 12, where m = the mean amount of vitamin C (in mg per serving) in beans frozen using the new method. You calculate that the power of the test against the alternative Ha :μ = 13.5 is 0.75. Which of the following is the best interpretation of this value? a. The complement of the probability of making a Type I error. b. The probability of concluding that the true mean is 12 mg/serving when it is actually 13.5 mg/serving. c. The probability of concluding that the true mean is higher than 12 mg/serving when it is actually 12 mg/serving. d. The probability of concluding that the true mean is 13.5 mg/serving when it actually 1

e. The probability of concluding that the true mean is higher than 12 mg/serving when it is actually 13.5] mg/serving.

A medical experiment compared the herb Echinacea with a placebo for preventing colds. The study used 50 different response variables usually associated with colds, such as low-grade fever, congestion, frequency of coughing, etc. The subjects were 40 women between 25 and 40 years of age. At the end of the study, the Echinacea group displayed significantly better responses at the a = 0.05 level for three of the 50 response variables studied. Which of the following is an appropriate conclusion to draw from this study? a. There is good evidence that Echinacea reduces cold symptoms b. There is good evidence that Echinacea reduces at least these three cold symptoms c. There is good evidence that Echinacea reduces cold symptoms, but we should be careful not to extend our conclusions beyond women in this age group d. There may be some benefits to Echinacea, but further study with a larger number of subject is necessary e. Th

e. There is not sufficient evidence to conclude that Echinacea is beneficial. It is quite likely that the significant results for the three variables occurred by chance

When are the results of a study statistically significant?

if the p-value (or prob of getting our sample if the bull hyp is true) is lower than alpha (the significance level)

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test Ho: μ=5 Ha: μ≠5 If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level against the alternative μ = 5.1 is 0.23. (c) If you decide to use α = 0.10 in place of α = 0.05, with no other changes in the test, will the power increase or decrease? Justify your answer.

increase in alpha will increase power

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test Ho: μ=5 Ha: μ≠5 If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level against the alternative μ = 5.1 is 0.23. (b) You could get higher power against the same alternative with the same by changing the number of measurements you make. Should you make more measurements or fewer to increase power?

increase measurements to increase power

For a mean P H A N T O M

one sample t test

For a Proportion P H A N T O M

p=the true proportion of all (context)

Error probabilities You read that a statistical test at the α = 0.01 level has probability 0.14 of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

power= 1-B=1-0.14=0.86

Give two explanations for why the sample proportion was below/above/ not equal to expected

random chance the true proportion of the context is actually below/above/ not equal to the null

Is the ratio of male births to female births even? A simple random sample of births in a major metropolitan area found 1345 boys among 2546 firstborn children. A 99% confidence interval for =the proportion of male births in this population is given by (0.5028, 0.5538). (a) Use the confidence interval to draw a conclusion about the hypothesis Ho :p = 0.5 against Ha: p ≠ 0.5. Be sure to indicate the appropriate significance level.

since the 99% CI doesn't contain 0.5 we have sufficant evidence to reject null ergo we support the true proportion of males birth is different from the proportion of female births We have sufficent evidence to reject the null ergo we can support the cliam that the ratio of male birth

14.A certain cigarette brand advertises that the mean nicotine content of their cigarettes is 1.5 mg, but you are suspicious and plan to investigate the advertised claim by testing the hypotheses Ho:μ = 1.5 versus Ha:μ > 1.5 at the α = 0.05 significance level. You will do so by measuring the nicotine content of 30 randomly selected cigarettes of this brand. (a) Describe what a Type I error would be in this context.

we would reject the mean context of their cigarettes is greater than 1 song when really the mean level is greater than 1.5mg

For a mean P H A N T O M

x̄- μ =t* (s/√n)

When the manufacturing process is working properly, Never Ready batteries have lifetimes that follow a slightly right-skewed distribution with μ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded. (a) Define the parameter of interest and state appropriate hypotheses for the quality control supervisor to test.

μ= the true mean lifetime of batteries produced by NeverReady Ho: μ=7 Ha: μ<7

For a mean P H A N T O M

μ= the true mean of all (context)

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test Ho: μ=5 Ha: μ≠5 If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level against the alternative μ = 5.1 is 0.23. d) If you shift your interest to the alternative μ = 5.2, with no other changes, will the power increase or decrease? Justify your answer.

μ=5-5.1=0.1 μ=5-5.2=0.2 increase the distance from the null the power will increase