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Week 7 Assignment Hypothesis
Test for the Mean - Population
Standard Deviation Known
Question
Jamie, a bowler, claims that her bowling score is less than 168 points, on average. Several of her teammates do not believe her, so she decides to do a hypothesis test, at a 1% significance level, to persuade them. She bowls 17 games. The mean score of the sample games is 155 points. Jamie knows from experience that the standard deviation for her bowling score is 19 points. H 0 : μ ≥168; Ha : μ < α =0.01 (significance level) What is the test statistic ( z -score) of this one-mean hypothesis test, rounded to two decimal places?
Question
A politician claims that at least 68 % of voters support a decrease in taxes. A group of researchers are trying to show that this is not the case. Identify the researchers' null hypothesis, H 0 , and the alternative hypothesis, Ha , in terms of the parameter p.
Question
Correct answer: A normal curve is over a horizontal axis and is centered on 14.7. Two ticks are labeled on the axis, one at a point greater than 1.47 and one at a point less than 1.47. The area under the curve to the left of the point less than 14.7 and right of the point greater than 14.7 is shaded. The alternative hypothesis, Ha , tells us which area of the graph we are interested in. Because the alternative hypothesis is X ≠14.7, we are interested in the region larger than or smaller than (to the left or the right of) 14.7, so the correct graph is the third answer choice.
Question
Which of the following results in a null hypothesis p =0.3 and alternative hypothesis p ≠0.3? Correct answer: An insurance company claims that 30 % of adults between the ages of 30 and 40 are overweight. A group of doctors think that is not accurate, and wants to show that the percent of these adults that are overweight is not 30 %. Remember that the null hypothesis, p =0.3, is the claim that the researchers (in this case, the doctors) are trying to reject. So the insurance company's claim should be that p =0.3 and the group of doctors should try to show that p ≠0.3. This is the fourth answer choice.
Question
Annie, a long jumper, claims that her jump distance is not equal to 19 feet, on average. Several of her teammates do not believe her, so she decides to do a hypothesis test, at a 10% significance level, to persuade them. She makes 24 jumps. The mean distance of the sample jumps is 19.5 feet. Annie knows from experience that the standard deviation for her jump distance is 1.2 feet. H 0 : μ =19; Ha : μ ≠ α =0.1 (significance level) What is the test statistic ( z -score) of this one-mean hypothesis test, rounded to two decimal places?
Question
Determine the Type II error if the null hypothesis, H 0 , is: a wooden ladder can withstand weights of 250 pounds and less.
Correct answer: A normal curve is over a horizontal axis and is centered on 6.4. A tick is labeled on a point to the left of 6.4. The area under the curve and to the left of the tick is shaded. The alternative hypothesis, Ha , tells us which area of the graph we are interested in. Because the alternative hypothesis is X <6.4, we are interested in the region less than (to the left of) 6.4, so the correct graph is the first answer choice.
Question
Which graph below corresponds to the following hypothesis test? H 0 : p ≤8.1, Ha : p >8.
Correct answer: A normal curve is over a horizontal axis and is centered on 8.1. A vertical line segment extends from the horizontal axis to the curve at a point to the right of 8.1. The area under the curve to the right of this point is shaded. The alternative hypothesis, Ha , tells us which area of the graph we are interested in. Because the alternative hypothesis is p >8.1, we are interested in the region greater than (to the right of) 8.1, so the correct graph is the second answer choice.
Question
Which graph below corresponds to the following hypothesis test? H 0 : μ ≤16.9, Ha : μ >16.
Correct answer: The researchers conclude that the rack holds less than 100 pounds of force, but the rack actually holds more than 100 pounds. Remember that a Type I error is rejecting the null hypothesis when the null hypothesis is true and a Type II error is not rejecting the null hypothesis when it is false. We are asked for the Type I error in this scenario. Rejecting the null hypothesis means rejecting the statement that the rack can hold more than 100 pounds. Therefore, a Type I error is: The researchers conclude that the rack holds less than 100 pounds when in reality it holds at least 100 pounds.
Question
Determine the Type I error if the null hypothesis, H 0 , is: the percentage of homes in the city that are not up to the current electric codes is no more than 10 %. And, the alternative hypothesis, Ha , is: the percentage of homes in the city that are not up to the current electric codes is more than 10 %. Correct answer: There is sufficient evidence to conclude that more than 10 % of homes in the city are not up to the current electrical codes when, in fact, there are no more than 10 % that are not up to the current electric codes. A Type I error is the decision to reject the null hypothesis when it is true. In this case, the Type I error is when there is sufficient evidence to conclude that more than 10 % of homes in the city are not up to the current electrical codes when, in fact, there are no more than 10 % not up to the current electric codes.
Question
Determine the Type I error if the null hypothesis, H 0 , is: a wooden ladder can withstand weights of 250 pounds and less. Correct answer: You think the ladder cannot withstand weight of 250 pounds and less when, in fact, it really can. A Type I error is the decision to reject the null hypothesis when it is true. In this case, the Type I error is thinking the wooden ladder cannot withstand the weights of 250 pounds or less when it really can.
Question
Which of the following results in a null hypothesis p ≤0.47 and alternative hypothesis p >0.47? Correct answer: An online article claims that at most 47 % of internet users participate in social media. A group of researchers think this is incorrect, and they want to show that more than 47 % of internet users participate in social media. Remember that the null hypothesis is the statement that the researchers are trying to reject, or show is wrong. In this case, the null hypothesis is p ≤0.47, which should be what the online article claims. The alternative hypothesis, p >0.47, should be what the researchers are trying to show. So the fourth answer choice is correct.
Question
A car magazine claims that 68 % of car owners follow a normal maintenance schedule. A mechanic does not think this is accurate, and so he wants to show that the percentage of people who follow a normal maintenance schedule is not equal to 68 %. Identify the null hypothesis, H 0 , and the alternative hypothesis, Ha , in terms of the parameter p.
The store claims that the mattresses last at least 5 years. In symbols, μ ≥ 5. This is the null hypothesis, H 0 , because this is the assumption the mattress store has been making. The consumer group is trying to reject this claim. The alternative hypothesis, Ha , is the opposite of the null hypothesis. So Ha : μ < 5.
Question
Which of the following results in a null hypothesis p ≤0.61 and alternative hypothesis p >0.61? Correct answer: A study says that at most 61 % of students study less than 5 hours per week. A researcher thinks this is incorrect, and wants to show that more than 61 % of students study less than 5 hours per week. Remember that the alternative hypothesis is the claim that the researcher is trying to show. The null hypothesis, p ≤0.61 corresponds to the claim in the study, and the alternative hypothesis, p >0.61 corresponds to what the researcher is trying to show (to reject the null hypothesis). So the third choice is the correct answer.
Question
Which of the following results in a null hypothesis p ≤0.69 and alternative hypothesis p >0.69? Correct answer: A mechanic wants to show that more than 69 % of car owners follow a normal maintenance schedule, contrary to a study that found that the percentage was at most 69 %. Consider each of the options. The scenario in option B has the null hypothesis p ≤0.69 based on the words at most and the fact that the null hypothesis is always stated with some form of equality. Also, remember that the alternative hypothesis is the statement that the mechanic is trying to show. In this case, the mechanic wants to show that more than 69 % of car owners follow a
normal maintenance schedule, so the alternative hypothesis, Ha is p >0.69. The null hypothesis, H 0 , is the opposite of that, p ≤0.69.
Question
Which of the following results in a null hypothesis μ ≥ 31 and alternative hypothesis μ < 31? Correct answer: A hospital claims that the mean wait time for emergency room patients is at least 31 minutes. A group of researchers think this is inaccurate and wants to show that the mean wait time is less than 31 minutes. The null hypothesis, μ ≥ 31 , is the claim that the researchers are trying to reject. In this case, the researchers are trying to reject the hospital's claim, so the hospital's claim should be μ ≥ 31. In words, they claim that the average wait time is at least 31. The alternative hypothesis is the claim that the researchers are trying to demonstrate, μ < 31. So the researchers want to show that the average wait time is less than 31 minutes. Thus, the third choice is correct.
Question
Which of the following results in a null hypothesis μ ≤7 and alternative hypothesis μ >7? Correct answer: A study wants to show that the mean number of hours of sleep the average person gets each day is more than 7. Consider each of the options. The scenario in the third option has the null hypothesis μ ≤ 7 based on the words "more than" and the fact that the null hypothesis is always stated with some form of equality. Also, remember that the alternative hypothesis is always the statement that is trying to be shown by the study.
Question
Question
Suppose the null hypothesis, H 0 , is: doctors believe that a surgical procedure is successful at least 80% of the time. Which of the following gives β , the probability of a Type II error? Correct answer: the probability that doctors think the surgical procedure is successful at least 80 % of the time when, in fact, it is not A Type II error is the decision not to reject the null hypothesis when, in fact, it is false. In this case, the Type II error is when the doctors think that the surgical procedure is successful at least 80 % of the time when, in fact, it is not.
Question
What is α , the probability of a Type I error if the null hypothesis, H 0 , is: Carmin believes that her chemistry exam will only cover material from chapters four and five. Correct answer: the probability that Carmin believes that her chemistry exam will not cover material only from chapters four and five when, in fact, it will only cover material from chapters four and five
A Type I error is the decision to reject the null hypothesis when it is true. In this case, the Type I error is when Carmin believes that her chemistry exam will not cover only material from chapters four and five when, in reality, it will.
Question
Suppose the null hypothesis, H 0 , is: a sporting goods store claims that at least 70% of its customers do not shop at any other sporting goods stores. What is the Type I error in this scenario? Correct answer: The sporting goods store thinks that less than 70 % of its customers do not shop at any other sporting goods stores when, in fact, at least 70 % of its customers do not shop at any other sporting goods stores. A Type I error is the decision to reject the null hypothesis when it is true. In this case, the Type I error is when the store thinks that less than 70 % of its customers only shop at their sporting goods store when, in fact, it is at least 70 %.
Question
Determine the Type II error if the null hypothesis, H 0 , is: researchers claim that 65% of college students will graduate with debt. Correct answer: The researchers think that 65 % of college students will graduate with debt when, in fact, more or less than 65 % of college students will graduate with debt. A Type II error is the decision not to reject the null hypothesis when, in fact, it is false. In this case, the Type II error is when the researchers think that 65 % of college students will graduate with debt when, in fact, more or less than 65 % will graduate with debt.
