Column 1
What is the 4-step process for a significance test
STATE
PLAN
DO
CONCLUDE
FILL IN THE BLANK
If the P-value is small, we _____ Ho and conclude there is convincing evidence for Ha
If the P-value is not small, we _______ Ho and conclude there is not convincing evidence for
REJECT
FAIL TO REJECT
FILL IN THE BLANK
A Type I error occurs if we _______ Ho when Ho is true. That is, the data give convincing evidence that Ha is true when it really isn’t.
A Type II error occurs if we _______ Ho when Ha is true. That is, the data do not give convincing evidence that Ha is true when it really is.
REJECT
FAIL TO REJECT
Which one is the Null hypothesis
Ha OR Ho
Ho
When interpreting a P-value, what word do you start the sentence with?
Assuming
Which of the following is not a required condition for performing a t-test about an unknown population mean
(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.
t-procedures use s, the sample standard deviation, to estimate σ, so we do not need to know σ to use them.
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.
t-procedures use s, the sample standard deviation, to estimate σ, so we do not need to know σ to use them.
Looking online, you find the salaries of all 25 players for the Chicago Cubs as of opening day of the 2013 baseball season. The club total was $104 million, fourteenth in the major leagues. Which inference procedure would you use to estimate the average salary of the Cubs players?
(a) One-sample z interval for µ
(b) One-sample t interval for µ
(c) One-sample t test
(d) One-sample z test
(e) None of these—this is not a situation that calls for inference
(e) None of these—this is not a situation that calls for inference
Since we already know the salary of every member of this population, we can easily calculate the parameter value. There is no need for statistical inference
Resting pulse rate is an important measure of the fitness of a person's cardiovascular system, with a lower rate indicative of greater fitness. The mean pulse rate for all adult males is approximately 72 beats per minute. A random sample of 25 male students currently enrolled in the Agriculture School at a major university was selected and the mean resting pulse rate was found to be 80 beats per minute with a standard deviation of 20 beats per minute. The experimenter wishes to test if the students are less fit, on average, than the general population.
Which of the following describes a Type II error in this setting?
(a) Concluding that the students are less fit (on average) than the general population when in fact they have equal fitness on average.
(b) Not concluding that the students are less fit (on average) as the general population when in fact they are less fit (on average).
(c) Not concluding that the students are less fit (on average) as the general population when in fact they have the same fitness (on average).
(d) Concluding that the students are less fit (on average) than the general population, when, in fact, they are less fit (on average).
(e) Concluding that the students have the same fitness (on average) when in fact they are more fit (on average).
(b) Not concluding that the students are less fit (on average) as the general population when in fact they are less fit (on average).
A Type II error is failing to reject a false Ho, which in this context means concluding that the mean pulse of the agriculture students is not above 72 when it is.
Which of the following describes a Type I error?
You are testing whether a new type of light bulb lasts longer than the standard one. The null hypothesis (H₀) is that the new bulb lasts the same as the standard bulb, and the alternative hypothesis (H₁) is that the new bulb lasts longer.
a) Concluding that the new bulb lasts longer when it actually does not.
b) Concluding that the new bulb does not last longer when it actually does.
c) Concluding that both bulbs last the same when the new one actually lasts longer.
d) Concluding that both bulbs last the same when they actually don't.
a) Concluding that the new bulb lasts longer when it actually does not.
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
Area under standard normal curve above z = 0.8 is 0.2119. Since the test is two-tailed, the P-value is 2×0.2119 =0.4238.
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.
The question asks if there is reason to think that more than half of all seniors think their job prospects are good, so we are interested in the one-tailed alternative above 0.5.
In a test of Ha: 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.
Since the P-value is greater than α, we have insufficient evidence against Ho, so we cannot conclude that Ho is false. The test only provides evidence against the null, not in support of the alternative.
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 H0: 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.
(b) Fail to reject Ho at the α = 0.05 level of significance.
Since the null value of 0 is in the 95% confidence interval, we cannot reject Ho.
A significance test was performed to test the null hypothesis Ho: p = 0.5 versus the alternative Ha: p>0.5. The test statistic is z = 1.40. Which of the following is closest to the P-value for this test?
(a) 0.0808
(b) 0.1492
(c) 0.1616
(d) 0.2984
(e) 0.9192
(a) 0.0808
Area under standard normal curve above z = 1.40 is 0.0808.
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 µ = 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 is actually 12 mg/serving.
(e) The probability of concluding that the true mean is higher than 12 mg/serving when it is actually 13.5 mg/serving
(e) The probability of concluding that the true mean is higher than 12 mg/serving when it is actually 13.5 mg/serving.
Power is the probability of rejecting a false null hypothesis, which in this case is concluding that the vitamin C content is higher than 12 mg/serving when it is indeed higher
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.
Power is the probability of rejecting a false null hypothesis, which in this case is concluding that the true mean daily Calcium intake for women a the college is lower than 1000 mg when it is actually 900 mg.
You collect test scores on four members of a population which you can safely assume is approximately Normally distributed and test the hypotheses Ho: µ = 100 versus Ha :µ > 100. You obtain a P-value of 0.052. Which of the following statements is true?
(a) At the 5% significance level, you have proved that H0 is true.
(b) You have failed to obtain any evidence for Ha.
(c) There is some evidence against H0, and a study using a larger sample size may be worthwhile.
(d) You can accept Ha at the 5% significance level.
(e) You can accept H0 at the 10% significance level
(c) There is some evidence against Ho, and a study using a larger sample size may be worthwhile
A test based on a sample of 4 individuals has very little power, so getting a P-value so close to 0.05 suggests that the likelihood of a Type II error is high. A larger sample would increase the power of the test
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 than 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 true proportion is 1/12.
(e) 0.0056 is the probability of getting a sample with a proportion of September birthdays this close to 1/12 if the true proportion is not 1/12.
(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.
The P-value measures the probability of getting a result as extreme as the sample statistic is when the null hypothesis is true.
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 gives the complete set of correct responses.
(b) I and III only
I is true because power is greater if the alternative (actual) value of the parameter is farther from the null. III is true because P(Type I error) = α and P(Type II error) depends on α.
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.
(b) Interpret the P-value from your test in the context of the problem.
(a) State: We wish to test Ho :p = 0.176 versus Ha : p < 0.176 where p = the true proportion of Thicksburg residents who are underemployed. We will use a significance level of α = 0.05.
Plan: The procedure is a one-sample z-test for a proportion.
Conditions: Random: The mayor’s staff took an SRS of 300 residents. 10%: We can safely assume that there are more than 10 300 3000 × = residents of Thicksburg
Do: P-value= 0.1190
Conclude: A P-value of 0.1190 is greater than α = 0.05, so we fail to reject Ho: there is insufficient evidence to conclude that the proportion of Thicksburg residents who are underemployed is below the national proportion.
(b) If the true proportion of underemployed residents is 0.176, the probability of getting a sample proportion of underemployed residents as far or farther below 0.176 as our sample is 0.1190.
When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with µ = 7hours. 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.
(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)
(a) Ha: µ = 7 versus Ha : µ < 7, where µ = the mean lifetime of batteries produced by this manufacturing process.
(b) This sample size is too small for a population that is known to be right-skewed.
When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with µ = 7hours. 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) Describe a Type I and a Type II error in this situation and the consequences of each.
(b) 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
(a) Type I error: Concluding that the mean lifetime is less than 7 hours when it is equal to (or more than) 7 hours. The quality control supervisor would throw out good batteries. Type II error: Not concluding that the mean lifetime is less than 7 hours when it is. The company will sell batteries that have a shorter lifetime than advertised.
(b) Lowering the significance level to 1% would decrease the probability of a Type I error, but increase the probability of a Type II error and decrease the power of the test. This would increase the likelihood of selling batteries with a short lifetime but reduce the number of batteries that are discarded.
A chemical company is developing a new fertilizer for tomatoes and wants to know if tomatoes grown with it are larger than the known mean weight of tomatoes grown with their current fertilizer, which is µ = 1.2 pounds. They will feed 30 young plants with the new fertilizer and measure the mean weight of fruits produced by the plants.
(a) State the null and alternative hypotheses for this test.
(b) Describe a Type II error in the context of this study and its possible consequences.
(a) Ho: µ = 1.2; Ha: µ > 1.2, where µ = the true mean weight of tomatoes grown with the new fertilizer.
(B) A Type II error would be concluding that tomatoes grown with the new fertilizer are not larger than those grown with the current fertilizer, when in fact they are larger.
The consequence would be that the company might abandon development of a potentially profitable product.
Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal means that the paper will live forever in libraries. The British Medical Journal combines the two: it prints short and readable versions, with longer versions available online. Is this OK with authors? The journal asked a random sample of 104 of its recent authors several questions. One question was “Should the journal continue using this system?” In the sample, 72 said “Yes.”
(a) Do the data give good evidence that more than two-thirds (67%) of authors support continuing this system? Carry out an appropriate test to help answer this question.
State: We wish to test Ho: p = 0.67 versus Ha: p > 0.67, where p = the true proportion authors who support the system of only publishing longer papers online. We will use a significance level of α = 0.05.
Plan: The procedure is a one-sample z-test for a proportion
Conditions: Random: The journal took a random sample of 104 recent authors. 10%: It seems reasonable to assume that there are more than 10 104 1040 × = recent authors of articles in the British Medical Journal.
Do: P-value = 0.3336
Conclude: A P-value of 0.3336 is greater than α = 0.05, so we fail to reject H0: there is insufficient evidence to conclude that the proportion of recent authors who support the system is greater than 67%.