M2209 Practice Midterm Solutions
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Mathematics
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Apr 3, 2024
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Math 2209 Practice Questions for the Midterm Question 1 To be selected to serve on a jury, one must be on the list of registered voters in the province and have reached the age of majority (which is 19 in Nova Scotia). The average age for registered voters in a province is 39.7 years, with a standard deviation of 11.8 years. a.
QUESTION 1
What shape do you think this distribution has? Explain. The distribution of the age of jurors will be skewed to the right. The age of jurors can’t be lower than 19 but has a higher limit on the right. [there may be outliers] b.
QUESTION 2 Can you use Table Z to find the probability that a randomly chosen juror is at least 50 years old? Explain. No, because the age of jurors does not follow a normal model. c.
If we were to take a random sample of 50 jurors, will the assumptions for the central limit theorem hold? Explain in the context of the question. QUESTION 3 Independence Assumption - Randomization condition: met, since we are told they are choosing jurors randomly from the population of all potential jurors in NS. 10% (big population) condition: met, since there are many adults in NS. As noted in part (a), the normal Population Assumption is not met. So we need to check the sample size assumption. QUESTION 4 Sample size assumption - Large enough sample condition: with n = 50, this is > 40 so large enough to assume that the sample means are approximately normal, regardless of the shape of the distribution of the age of jurors. d.
QUESTION 5 What are the mean and standard deviation for the average age of jurors in a sample of 50 jurors? Mean (
𝝁𝝁
𝒚𝒚
�
) = 39.7 Standard deviation (SD(
𝒚𝒚
�
)) = 𝝈𝝈 √𝒏𝒏
⁄
= 11.8/
√𝟓𝟓𝟓𝟓
= 1.669 e.
QUESTION 6
Using the 68-95-99.7 rule, there is a 0.95 probability that the average age of these 50 jurors falls between what two values? 39.7 – 1.669(2) = 36.362 39.7 + 1.669(2) = 43.038 (36.362, 43.038)
Question 2 A random sample of 15 hamburgers, selected from all those produced by a fast food restaurant during one day, were measured for fat content. The average fat content of these 15 hamburgers was 1.85 mg, the standard deviation was 0.29 mg. Is there evidence to believe that the average fat content of all the hamburgers produced by this fast food restaurant on that day was less than 2 mg? Answer this question by following the steps below. There is a Minitab output in the data file to help you. a)
QUESTION 7
State the hypotheses to be tested. H
0
: µ = 2 H
A
: µ < 2 b)
QUESTION 8 Identify the test statistic and p-value from the Minitab output. t = -2, p = 0.032 c)
QUESTION 9 Briefly assess the strength of the evidence. Moderate evidence for Ha d)
QUESTION 10
State your conclusion in the context of the problem. Since the assumptions are met, we have moderate evidence (0.032 < 0.05) that the population average fat content of all the hamburgers produced by this fast food restaurant on that day was less than 2 mg.
Question 3 An accountant wishes to determine how closely tax assessors and real estate agents agree on the value of a home. A random sample of 12 tax assessors were asked to appraise the home, and similarly, a random sample of 20 real estate agents were also asked to appraise the home. The assumptions and conditions were checked and are all met, and the results of their appraisals (in thousands of dollars) are summarized in the Minitab Output in the data file. a.
QUESTION 11 Is the independent group assumption met? Explain in context. Note that the other assumptions were checked and are met. Yes, since they took a random sample of tax assessors and a separate random sample of real estate agents (OR: yes, as long as someone can’t be both a tax assessor and real estate agent) b.
QUESTION 12 If you needed to calculate the 98% confidence interval, what would be the degrees of freedom and t* that you would use? DF = 27 and t* = 2.473 from Table T. c.
QUESTION 13 Interpret the interval, in context (you will find the interval on the Minitab output). Since the assumptions hold, we are 98% confidence that the population mean appraised values by tax assessors is between 4.52 and 14.28 thousand dollars higher than the appraised value by real estate agents. [OR: Since the assumptions hold, we are 98% confident that the difference between the mean appraised values for populations of all tax assessors and all real estate agents is between 4.52 and 14.28 thousand dollars.] d.
QUESTION 14 Based on the confidence interval, is there any evidence of a difference between what tax assessors say and what real estate agents say? Explain. Yes, the interval shows a difference since 0 is not in the range.
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Question 4 The violent crime rate (number of violent crimes per 100,000 residents) is investigated for nine randomly selected Canadian cities for the years 1990 and 2000 to see if there has been a change. Follow the steps below to answer this question. The conditions were checked, and are met. a.
QUESTION 15 State hypotheses, defining the parameter used in context. H
0
: µ
d
= 0 H
A
: µ
d
≠ 0 where µ d is the average difference (or change) in the violent crime rate of the population of all cities in Canada between 2000 and 1990. b.
QUESTION 16
Identify the test statistic and p-value from the Minitab output. t = -5.01 , p=value = 0.001 c.
QUESTION 17 Using the p-value from the Minitab output, briefly assess the strength of the evidence. Strong evidence for Ha d.
QUESTION 18 State your conclusion in context. Since the assumptions are met, we have strong evidence that there has been a change in the average crime rate between 1990 and 2000, for the population of all Canadian cities.
Question 5 A television executive is planning sports programming for next year. She has information about viewers' preferences from previous years (historical proportions, shown in the Test Proportion column). A random sample of 620 U.S. adults was asked to name their favourite sport. Is there evidence at the 5% level of significance that the distribution has changed from the previous years? Follow the steps below to answer this question. The Minitab output is in the data file. a)
QUESTION 19
Write Ho and Ha in plain English, including the population. Ho: The distribution of Favorite sport has not changed from the previous year, for the population of all U.S. adults. Ha: The distribution of Favorite sport has changed from the previous year, for the population of all U.S. adults. b)
QUESTION 20 State test statistic and p-value from the output. Chi-Sq = 24.1683 p-value = 0.002 c)
QUESTION 21 Briefly assess the significance of the evidence. 0.002 < 0.05, reject Ho and conclude Ha. d)
QUESTION 22 Give a full conclusion in context. Since the assumptions are met, we can conclude that the distribution of favorite sport has changed from the previous year, for the population of all U.S. adults. e)
QUESTION 23 What type of error (Type I or Type II) could you have made here? Explain in context.
Type I. We concluded the distribution of favorite sport has changed, when maybe it has not.
Question 6 A car insurance company performed a study to determine whether an association exists between age and the frequency of car accidents. They randomly selected 300 claims, and the results are in the Data file. a.
QUESTION 24 Is this a test of homogeneity or independence? Explain in the context of this question. This is a test of independence, since there is one population of claims and two categorical variables, age and number of accidents. b.
QUESTION 25
State the Expected Cell Frequency condition and explain whether or not this condition is met for this question. Condition: the expected count for each cell needs to be greater than 5.
Is it met? No, because 3 of the cells have an expected count of 4.33 and 4.33 < 5.
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