alpha, beta and power busmgt2320

consider the test of Ho: defendant is not guilty Ha: defendant is guilty a. explain in context the conclusion of the test if Ho is rejected b. describe the consequence of a type 1 error c. explain in context the conclusion of the test if you fail to reject Ho d. describe the consequence of a type 2 error

a. rejecting Ho implies that the null, the defendant is not guilty, is false. the defendant is therefore guilty and sentenced b. type 1 would result in an innocent defendant being convicted for a crime they didn't commit c. failing to reject Ho implies that the null, the defendant is not guilty, is a possibility. the defendant is acquitted and set free d. a type 2 error would result in a guilt defendant being set free and not being convicted

what is the level of significance of a test of hypothesis?

the level of significance, a, is the probability of committing a type 1 error, rejecting the null when it is in fact true

Describe type I and type II errors for a hypothesis test of the indicated claim. A clothing store claims that no more than 10​% of its new customers will return to buy their next article of clothing.

type 1 error: A type I error will occur when the actual proportion of new customers who return to buy their next A article of clothing is no more than 0.10​, but you reject Upper H 0​: p <=0.10. type 2 error: A type II error will occur when the actual proportion of new customers who return to buy their next A article of clothing is more than 0.10​, but you fail to reject H0​: p<=0.10.

state whether a Type​ I, a Type​ II, or neither error has been made A pharmaceutical company tests whether a drug lifts the headache relief rate from the​ 25% achieved by the placebo. They fail to reject the null hypothesis because the​ P-value is 0.465. Further testing shows that the drug actually relieves headaches in​ 38% of people. Choose the correct answer below.

The company made a Type II error. The null hypothesis was not​ rejected, but it was false. The true relief rate was greater than 0.25.

Suppose you want to estimate the proportion of traditional college students on your campus who own their own car. You have no preconceived idea of what that proportion might be. What sample size is needed if you wish to be 90​% confident that your estimate is within 0.04 of the true​ proportion?

423 ME=z*square root(p-hat x q hat) / n) round z* to three decimal places

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Unknown to the statistical​ analyst, the null hypothesis is actually true. A. If the null hypothesis is rejected a Type I error would be committed. Your answer is correct.B. If the null hypothesis is rejected a Type II error would be committed. C. If the null hypothesis is not rejected a Type I error would be committed. D. If the null hypothesis is not rejected a Type II error would be committed. E. No error is made.

A

A marketing analyst at an internet book store is testing a new web design which she hopes will increase sales. She wants to randomly send n customers to the new site.​ She's hoping for an increase of​ 10% in sales from the new site. She calculates a power of 0.43 for that increase from a sample of 150 customers. Explain to her why she should consider a larger sample size. A. A larger sample size would decrease the​ power, which is the probability that a test will detect a false hypothesis. B. A larger sample size would decrease the standard​ deviation, and making the sample standard deviation smaller increases the power without changing the alpha level. C. A power of 0.43 means that she will correctly reject the null hypothesis of no increase less than half the time. She should use a larger sample size to increase the probability that she will come to the correct decision. D. A power of 0.43 means that she will correctly reject the alternative hypothesis of an increase less than half the time. She should use a larger sample size to increase the probability that she will come to the correct decision.

B.

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Complete parts a through f below. The statistical analyst rejects the null hypothesis. A. If the null hypothesis is true a Type II error would be committed. B. If the null hypothesis is not true a Type II error would be committed. C. If the null hypothesis is true a Type I error would be committed. Your answer is correct.D. If the null hypothesis is not true a Type I error would be committed. E. No error is made.

C.

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Complete parts a through f below. Unknown to the statistical​ analyst, the null hypothesis is actually false. A. If the null hypothesis is rejected a Type II error would be committed. B. If the null hypothesis is rejected a Type I error would be committed. C. If the null hypothesis is not rejected a Type II error would be committed. Your answer is correct.D. If the null hypothesis is not rejected a Type I error would be committed. E. No error is made.

C.

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Complete parts a through f below. Unknown to the statistical​ analyst, the null hypothesis is actually false and the analyst rejects the null hypothesis. A. A Type II error has been committed. B. A Type I error has been committed. C. Both a Type I error and a Type II error have been committed. D. No error is made.

D

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Complete parts a through f below. Unknown to the statistical​ analyst, the null hypothesis is actually true and the analyst fails to reject the null hypothesis. A. A Type I error has been committed. B. Both a Type I error and a Type II error have been committed. C. A Type II error has been committed. D. No error is made.

D

For each of the following​ scenarios, indicate which type of statistical error could have been committed​ or, alternatively, that no statistical error was made. Complete parts a through f below. The statistical analyst fails to reject the null hypothesis. A. If the null hypothesis is true a Type I error would be committed. B. If the null hypothesis is not true a Type I error would be committed. C. If the null hypothesis is true a Type II error would be committed. D. If the null hypothesis is not true a Type II error would be committed. Your answer is correct.E.

D.

state whether a type 1, type 2 or neither error has been made A human resource analyst wants to know if the applicants this year​ score, on​ average, higher on their placement exam than the 52.5 points the candidates averaged last year. She samples 50 recent tests and finds the average to be 54.1 points. She fails to reject the null hypothesis that the mean is 52.5 points. At the end of the​ year, they find that the candidates this year had a mean of 55.3 points.

The analyst made a Type II error. The actual value was 55.3Th ​ points, which is greater than 52.5.

state whether a type 1, type 2 or neither error has been made a bank wants to know if the enrollment on their website is above 30% based on a small sample of customers. they test Ho: p =0.3 vs Ha: p>0.3 and reject the null. later they find out that actually 28% of all customers enrolled

The bank made a Type I error. The actual value is not greater than 0.3 but they rejected the null hypothesis.Th

state whether a type 1, type 2 or neither error has been made A student tests 100 students to determine whether other students on her campus prefer soda brand A or soda brand B and finds no evidence that preference for brand A is not 0.5.​ Later, a marketing company tests all students on campus and finds no difference. Choose the correct answer below.

The student did not make an error. The actual value is​ 0.50, which was not rejected.

A restaurant in a certain state is under new management. While under the previous​ management, 15​% of the customers rated their dining experiences as​ "unsatisfactory." The​ restaurant's current managers would like to test their belief that this percentage is now lower due to new training procedures. Suppose 73 customers were randomly chosen and asked to rate their dining experiences. Using alpha=0.05​, complete parts a through c below. a. Explain how Type I and Type II errors can occur in this hypothesis test. b. Calculate the probability of a Type II error occurring if the actual proportion of unsatisfied customers is 11​%. c. Calculate the probability of a Type II error occurring if the actual proportion of unsatisfied customers is 5​%.

a. A Type I error can occur if the proportion of unsatisfied customers is above or equal to A 0.15 and the null hypothesis is rejected. A Type II error can occur if the proportion of unsatisfied customers is below 0.15 and the null hypothesis is not rejected. b. .782 c. .109

The director of a state agency believes that the average starting salary for clerical employees in the state is less than ​$27,000 per year. To test her​ hypothesis, she has collected a simple random sample of 100 starting clerical salaries from across the state and found that the sample mean is ​$26,900. a. State the appropriate null and alternative hypotheses. b. Assuming the population standard deviation is known to be ​$1 comma 500 and the significance level for the test is to be 0.05​, what is the critical value​ (stated in​ dollars)? c. Referring to your answer in part​ b, what conclusion should be reached with respect to the null​ hypothesis? d. Referring to your answer in part​ c, which of the two statistical errors might have been made in this​ case? Explain.

a. Ho: u>= 27,000 Ha: U<27,000 b. 26,753.27 c. do not reject the null d. a type 2 error might have been made because the null was not rejected

Contact rate is defined as the percentage of plate appearances by a baseball player that the ball is put in play​ (as opposed to striking​ out). Historically, the average has been 80​%. Suppose one wants to test that this rate has not changed during the last season by randomly selecting 260 plate appearances by players and recording the percentage of players that put the ball in play. Using alpha=0.05​, complete parts a through c below. a. Explain how Type I and Type II errors can occur in this hypothesis. b. Calculate the probability of a Type II error occurring if the actual contact rate is 77​%. c. Calculate the probability of a Type II error occurring if the actual contact rate is 88​%.

a. A Type I error can occur if the proportion of plate appearances that result in the ball being put in play is equal to 0.80 and the null hypothesis is rejected. A Type II error can occur if the proportion of plate appearances that result in the ball being put in the play is not equal to 0.80 and the null hypothesis is not rejected. b. .7598 c. 0594

According to a​ survey, 22​% of households relied solely on their cell phones for phone service instead of landlines. It was also reported that this percentage has steadily increased over the previous years. A phone company would like to sample 126 households randomly to test the hypothesis that the proportion of cell​ phone-only households increased. Complete parts a through d below. a. Explain how Type I and Type II errors can occur in this hypothesis test. b. Using alpha=0.05​, calculate the probability of a Type II error occurring if the actual proportion of cell​ phone-only households is 0.31. c. Using alpha=0.01​, calculate the probability of a Type II error occurring if the actual proportion of cell​ phone-only households is 0.31. d. explain the differences in the results calculated in parts b and c

a. A Type I error can occur when the researcher concludes the average percentage of households relying on a cell phone increased, but the percentage did not increase. A Type II error can occur when the researcher concludes that the percentage of households relying on a cell phone did not increase, ​when, in​ fact, the percentage of households relying on a cell phone increased. b. .2389 c. .4602 d. when alpha increased, the probability of committing a type 2 error decreases

Public health officials believe that 98.3​% of children have been vaccinated against measles. A random survey found that 97​% of children have been vaccinated against measles. An analyst finds a​ 98% confidence interval for the true proportion of vaccinated children to be (0.9673, 0.9737) a) Explain why she can reject the null hypothesis that p=0.983 vs. =<0.983 at alpha=.01 A. The value of 0.97 lies within the interval which leaves​ 98% of the plausible values in the interval. So she can reject the null hypothesis that p=0.983. B. A​ 98% confidence interval leaves​ 2% on one side of the​ interval, so it can be used for a​ two-sided test at the alpha=0.01 level. If the hypothesized value falls outside a​ 98% confidence​ interval, it will have a​ p-value less than 0.01.​ Thus, she can reject the null hypothesis. C. The value of 0.983 lies above the interval which leaves​ 1% of the plausible values on that side. So she can reject the null hypothesis that p=0.983. D. A​ 98% confidence interval leaves​ 1% on each side of the​ interval, so it can be used for a​ one-sided test at the alpha =0.01 level. If the hypothesized value falls outside a​ 98% confidence​ interval, it will have a​ p-value less than 0.01.​ Thus, she can reject the null hypothesis. ​b) Explain why the difference may or may not be important. A. The difference is not important because the difference between the true value and the predicted value is so small. B. The true value may be as high as 97.37​%. She would need to do a cost analysis to determine the cost of​ pro-vaccination advertisting in order to increase measles vaccinations by 0.93​%. C. A confidence interval of only​ 98% is not high enough in regards to something as serious as a fatal disease amongst children. D. The true value may be as high as 98.3​%. She would need to do a cost analysis to determine the cost of​ pro-vaccination advertisting in order to increase measles vaccinations by 0.93​%.

a. D b. B

State regulators are checking up on repair shops to see if they are certifying vehicles that do not meet pollution standards. ​a) In this​ context, what is meant by the power of the test the regulators are​ conducting? ​b) Will the power be greater if they test 20 or 30 ​cars? Why? ​c) Will the power be greater if they use a 5​% or a 10​% level of​ significance? Why? ​d) Will the power be greater if the repair​ shop's inspectors are only a little out of compliance or a​ lot? Why?

a. The power of the test is the probability of detecting that the shop is not meeting standards when it is not. b. The power will be greater if they test 30 cars because a larger sample size increases the power of the test. c. The power will be greater with a 10​% level of significance because there will be more of a chance of rejecting H0. d. The power will be greater if the inspectors are out of compliance by a lot because larger problems are easier to detect.

An enthusiastic junior executive has run a test of his new marketing program. He reports that it resulted in a​ "significant" increase in sales. A footnote on his report explains that he used an alpha level of 6.4​% for his test.​ Presumably, he performed a hypothesis test against the null hypothesis of no change in sales. ​a) If instead he had used an alpha level of 5​%, is it more or less likely that he would have rejected his null​ hypothesis? Explain. ​b) If he chose the alpha level 6.4​% so that he could claim statistical​ significance, explain why this is not an ethical use of statistics.

a. it is less likely that he would reject his null bc lowering alpha decreases the chance of rejecting the null b. alpha levels must be chosen before examining the data. otherwise, the alpha level could always be selected to reject the null

Public health officials believe that 90.7​% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found​ that, among more than​ 13,000 children, only 88.3​% had been vaccinated. A statistician would reject the​ 90% hypothesis with a​ P-value of P=0.008. a. explain what the p-value means in context b. the result is statistically significant, but is it important? A. We conclude that the actual percentage of vaccinated children is below 90.7​% and is about 88.3​%. This drop is not important because only a​ 5% change or more can be considered important. B. We concluded that the actual percentage of vaccinated children is below 90.7​%. A 2.4​% drop would probably not be considered noteworthy but in​ context, if​ 1,000,000 children are vaccinated each year a 2.4​% difference accounts for 24000 more children not being​ vaccinated, which is important. C. A 2.4​% difference in child vaccinations in not important. D. All statistically significant results are important.

a. there is only a .8% chance that 90.7% is not the actual % of children vaccinated b. B

The average retirement age for a certain country was reported to be 57.3 years according to an international group dedicated to promoting trade and economic growth. With the pension system operating with a​ deficit, a bill was introduced by the government during the summer to raise the minimum retirement age from 60 to 62. Suppose a survey of 44 retiring citizens is taken to investigate whether the new bill has raised the average age at which people actually retire. Assume the standard deviation of the retirement age is 6 years. Using alpha=0.05​, answer parts a through c below. a. Explain how Type I and Type II errors can occur in this hypothesis test. b. Calculate the probability of a Type II error occurring if the actual population age is 58.6 years old. c. Calculate the probability of a Type II error occurring if the actual population age is 60.1 years old.

a. type 1 can occur when the researcher concludes the average retirement age increased, but the average retirement age did not increase. a type 2 error occur when the researcher concludes that the average retirement age did not increase when it actually did increase b. .583 c. .074

when we test Ho: u=0 vs Ha: U>0, we get a p-value of 0.05 1. what would the decision be for a significance level of .025? a. we would decide not to reject the null. we would have enough evidence that the pop mean is different from 0 b. we would decide to reject the null. we would not have enough evidence that the pop mean is different from 0 c. we would decide not to reject the null. we would not have enough evidence that the population mean is different from 0 d. we would decide to reject the null. we would have sufficient evidence that the pop mean is different from 0 2. if the decision is an error, what type of error is it? 3. suppose the sig level were instead .10. what decision would you make and if it is in error, what type of error is it? a. if the sig level were instead 10, we would decide to reject the null. f this decision were in error, type 1 b. if the sig level were instead .10, we would decide not to reject the null. if this decision were in error, it would be a type 2 c. if the sig level were instead a .10, we would decide to reject the null. if this decision were in error, it would be a type 2 d. if the sig level were instead .10, we would decide not to reject the null. if this decision were in error, it would be a type 1

c. type 2 a.