# Mathematics

9)

 In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 17 recent loans is taken. The average calculated from this sample is 4.85%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.5%. Compute 90% and 95% confidence intervals for the population mean 30-year fixed mortgage rate. Use Table 1. (Round intermediate calculations to 4 decimal places, “z” value and final answers to 2 decimal places. Enter your answers as percentages, not decimals.)
 Confidence Level Confidence Interval 90% % to % 95% % to

10)

 Consider the following hypotheses: H0: p ≥ 0.37 HA: p < 0.37 Which of the following sample information enables us to reject the null hypothesis at α = 0.05 and at α = 0.10? Use Table 1.
 α = 0.05 α = 0.10 a. x = 33; n = 100 b. x = 80; n = 285 c. = 0.34; n = 58 d. = 0.34; n = 416

a. Reject or Do not reject

11)

 In a multiple regression with two explanatory variables and 117 observations, it is found that SSR = 4.51 and SST = 8.86.
 a. Calculate the standard error of the estimate. (Round your answer to 2 decimal places.)
 se
 b. Calculate the coefficient of determination R2. (Round your answer to 4 decimal places.)
 R2

12)

 A retailer is looking to evaluate its customer service. Management has determined that if the retailer wants to stay competitive, then it will have to have at least a 91% satisfaction rate among its customers. Management will take corrective actions if the satisfaction rate falls below 91%. A survey of 1,450 customers showed that 1,305 were satisfied with their customer service. Use Table 1.
a. Select the hypotheses to test if the retailer needs to improve its services.
 H0: p = 0.91; HA: p ≠ 0.91 H0: p ≥ 0.91; HA: p < 0.91 H0: p ≤ 0.91; HA: p > 0.91
 b. What is the value of the appropriate test statistic? (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)
 Test statistic
 c. Compute the p-value. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
 p-value
d. What is the conclusion?
 The management will take corrective action. The management will not take corrective action.

13)

 Consider the following hypotheses:
 H0: μ = 360 HA: μ ≠ 360
 The population is normally distributed with a population standard deviation of 73. Use Table 1.
 a. Use a 10% level of significance to determine the critical value(s) of the test. (Round your answer to 2 decimal places.)
 Critical value(s) ±
 b-1. Calculate the value of the test statistic with = 389 and n = 80. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)
 Test statistic
b-2. What is the conclusion at α = 0.10?
 Do not reject H0 since the value of the test statistic is smaller than the critical value. Do not reject H0 since the value of the test statistic is greater than the critical value. Reject H0 since the value of the test statistic is smaller than the critical value. Reject H0 since the value of the test statistic is greater than the critical value.
 c. Use a 5% level of significance to determine the critical value(s) of the test. (Round your answer to 2 decimal places.)
 Critical value(s) ±
 d-1. Calculate the value of the test statistic with = 335 and n = 80. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)
 Test statistic
d-2. What is the conclusion at α = 0.05?
 Reject H0 since the value of the test statistic is not less than the negative critical value. Reject H0 since the value of the test statistic is less than the negative critical value. Do not reject H0 since the value of the test statistic is not less than the negative critical value. Do not reject H0 since the value of the test statistic is less than the negative critical value.

14)

 Use computer) Assume that X is a hypergeometric random variable with N = 55, S = 18, and n = 14. Calculate the following probabilities. (Round your answers to 4 decimal places.)
 a. P(X = 8) b. P(X ≥ 2) c. P(X ≤ 4)

15)

 For a sample of 41 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in \$1,000s). The regression results are shown.
 ANOVA df SS MS F Significance F Regression 2 3,519,512 1,759,756.0 10.29454 5.9E-04 Residual 24 4,102,577 170,940.7 Total 26 7,622,089
 a. Calculate the standard error of the estimate. (Round your answer to 2 decimal places.)
 b-1. What proportion of the variability in crime rate is explained by the variability in the explanatory variables? (Round your answer to 4 decimal places.)
 Explained proportion
 b-2. What proportion is unexplained? (Round your answer to 4 decimal places.)
 Unexplained proportion

16)

 Consider the following simple linear regression results based on 20 observations. Use Table 2.
 Coefficients Standard Error t Stat p-value Lower 95% Upper 95% Intercept 33.8605 4.4081 7.6814 0.0000 24.60 43.12 x1 0.3634 0.1204 3.0183 0.0074 0.11 0.62
a-1. Choose the hypotheses to determine if the intercept differs from zero.
 H0: β0 ≥ 0; HA: β0 < 0 H0: β0 ≤ 0; HA: β0 > 0 H0: β0 = 0; HA: β0 ≠ 0
a-2. At the 5% significance level, what is the conclusion to the hypothesis test? Does the intercept differ from zero?
 Do not reject H0 the intercept is greater than zero. Reject H0 the intercept is greater than zero. Reject H0 the intercept differs from zero. Do not reject H0 the intercept differs from zero.
 b-1. Construct the 95% confidence interval for the slope coefficient. (Negative values should be indicated by a minus sign. Round your intermediate calculations to 4 decimal places,”tα/2,df” value to 3 decimal places and final answers to 2 decimal places.)
 Confidence interval to
b-2. At the 5% significance level, does the slope differ from zero?
 Yes, since the interval does not contain zero. No, since the interval contains zero. Yes, since the interval contains zero. No, since the interval does not contain zero.

17)

 Consider the following hypotheses:
 H0: μ = 33 HA: μ ≠ 33
 The population is normally distributed. A sample produces the following observations:
 38 31 34 36 33 38 28
 Use the p-value approach to conduct the test at a 5% level of significance. Use Table 2.
 a. Find the mean and the standard deviation. (Round intermediate calculations to 4 decimal places. Round your answers to 2 decimal places.)
 Mean Standard deviation
 b. Calculate the value of the test statistic. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)
 Test statistic
c. Approximate the p-value of the test statistic.
 0.05 < p-value < 0.10 p-value > 0.20 0.10 < p-value < 0.20
d. What is the conclusion?
 Reject H0 since the p-value is greater than α. Reject H0 since the p-value is smaller than α. Do not reject H0 since the p-value is greater than α. Do not reject H0 since the p-value is smaller than α.

18)

 Let P(A) = 0.59, P(B) = 0.24, and P(A ∩ B) = 0.14.
 a. Calculate P(A | B). (Round your answer to 2 decimal places.)
 P(A | B)
 b. Calculate P(A U B). (Round your answer to 2 decimal places.)
 P(A U B)
 c. Calculate P((A U B)c). (Round your answer to 2 decimal places.)
 P((A U B)c)

19)

 The time required to assemble an electronic component is normally distributed with a mean and a standard deviation of 34 minutes and 20 minutes, respectively. Use Table 1.
 a. Find the probability that a randomly picked assembly takes between 26 and 40 minutes. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
 Probability
 b. It is unusual for the assembly time to be above 56 minutes or below 13 minutes. What proportion of assembly times fall in these unusual categories? (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
 Proportion of assembly times

20)

 Christine has always been weak in mathematics. Based on her performance prior to the final exam in Calculus, there is a 55% chance that she will fail the course if she does not have a tutor. With a tutor, her probability of failing decreases to 25%. There is only a 65% chance that she will find a tutor at such short notice.
 a. What is the probability that Christine fails the course? (Round your answer to 4 decimal places.)
 Probability
 b. Christine ends up failing the course. What is the probability that she had found a tutor? (Round your answer to 4 decimal places.)
 Probability

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