Math 124 Exam 2                                                      Name: ______________________

 

Short Answer: (Each part is worth 3 points).

  1. A calculus instructor is interested in finding the strength of a relationship between the final exam grades of students enrolled in Calculus I and Calculus II at his college.  The data (in percentages) are listed below.

 

Calculus I (x)      88     78        62        75        95        91        83        86        98

Calculus II (y)    81     80        55        78        90        90        81        80        100           

 

(a)    Graph a scatter plot of the data.

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                                               

 

(b)   Find an equation of the regression line.                (c)  Find the Pearson Correlation Coefficient.

 

y = 1.044x – 5.989                                                      r = 0.945

 

 

(c)    Predict a Calculus II exam score for a student who received an 80 in Calculus I.

 

Plug x = 80 into the equation above.

                        y = 1.044 (80) – 5.989 = 77.531

 

 

 

  1. The analytic scores on a standardized aptitude test are known to be normally distributed with mean μ = 610 and standard deviation σ = 115.

(a)    Draw a normal curve with the parameters labeled.

 

 

 

 

                                                                                                                                   

 

(b)   Shade the region that represents the proportion of test takers who scored less than 725.

 

(c)    Suppose that the area under the normal curve to the left of X = 725 is 0.8413.  Provide two interpretations of this result.

The proportion of people scoring less or the probability that someone randomly chosen scored less is 84.13%. 

  1. The board of examiners that administers the real estate broker’s examination in a certain state found that the mean score on the test was 526 and the standard deviation was 72.  If the board wants to set the passing score so that only the best 10% of all applicants pass, what will be the passing score?  Assume that the scores are normally distributed.

 

 

 

X = 618.16

 

 

 

 

 

 

Use the scatter diagrams shown, labeled a through f to solve the following problems: (3 points each).

  1. In which scatter diagram is r = 0.01?

 

 

 

e

 

 

 

 

  1. In which scatter diagram is r = 1?

 

 

 

b

 

 

 

 

  1. In which scatter diagram is r = -1?

 

 

 

 

a

 

 

 

 

 

 

Multiple Choice.  Circle the correct answer: (3 points each).

  1. Find the area under the standard normal curve between z = 0 and z = 3.

(a)    0.5000                         (b)  0.9987                         (c)  0.4987                         (d) 0.4641

 

  1. Find the area under the standard normal curve to the left of z = 1.5.

(a) 0.9332                         (b)  0.0668                         (c)  0.5199                         (d) 0.7612

 

  1. Find the area under the standard normal curve to the right of z = -1.25.

(a)    0.7193                         (b)  0.5843                         (c)  0.1056                         (d) 0.8944

 

  1. For a standard normal curve, find the z-score that separates the bottom 90% from the top 10%.

(a)  1.28                             (b)  0.28                             (c)  1.52                             (d) 2.81

 

  1. IQ test scores are normally distributed with a mean of 97 and a standard deviation of 11.  An individual’s IQ score is found to be 111.  Find the z-score corresponding to this value.

(a)    1.03                             (b)  –1.03                           (c)  1.27                             (d) –1.27

 

  1. Use the standard normal distribution to find P(0 < z < 2.25).

(a)    0.5122                         (b)  0.4878                         (c)  0.8817                         (d) 0.7888

 

  1. Suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000 and standard deviation $900.  Find the percentage of buyers who paid less than $148,200.

(a)  2.28%                          (b)  16.35%                        (c)  83.65%                        (d) 97.72%

 

  1. The amounts a group of 4 students paid for textbooks this semester had a mean of $235 and a standard deviation of $38.4.  Using a sample size of 3, find the mean and standard deviation of the sampling distribution.

(a)    ;       (b)  ;     (c)  ;    (d) ;

 

  1. The number of violent crimes committed in a day possesses a distribution with a mean of 4.1 crimes per day and a standard deviation of 6 crimes per day.  A random sample of 100 days was observed, and the sample mean number of crimes for the sample was calculated.  Describe the sampling distribution of the sample mean.

(a)    Approximately normal with mean = 4.1 and standard deviation = 6

(b)   Approximately normal with mean = 4.1 and standard deviation = 0.6

(c)    Shape unknown with mean = 4.1 and standard deviation = 6

(d)   Shape unknown with mean = 4.1 and standard deviation = 0.6

 

  1. Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses.  Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund.  A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 71.2, 51.6, 244.8, 497, 113.7, 166.2, 100.6, and 221.6.  What value will be used as the point estimate for the mean endowment of all private colleges in the United States?

(a)  8                                  (b)  1466.7                         (c)  209.529                       (d) 183.338

 

  1. The lengths of pregnancies are normally distributed with a mean of 264 days and a standard deviation of 20 days.  If 64 women are randomly selected, find the probability that they have a mean pregnancy between 264 days and 266 days.

(a)    0.5517                         (b)  0.7881                         (c)  0.2119                         (d) 0.2881

 

  1. A random sample of 40 students has a mean annual earnings of $3120 and a population standard deviation of $677.  Construct the 95% confidence interval for the population mean, μ.

(a)    $110 < μ < $210      (b)  $2910 < μ < $3330     (c)  $4812 < μ < $5342     (d) $1987 < μ < $2346

 

  1. Suppose a 95% confidence interval for μ turns out to be 120 < μ < 310.  To make more useful inferences from the data, it is desired to reduce the width of the confidence interval.  Which of the following will result in a reduced interval width?

(a)    Increase the sample size

(b)   Increase the sample size and decrease the confidence level.

(c)    Decrease the confidence level.

(d)   All of the choices will result in a reduced interval width.


 

  1. Suppose a 90% confidence interval for μ turns out to be 110 < μ < 260.  Based on the interval, do you believe the average is equal to 270?

(a)    Yes, and I am 90% sure of it.

(b)   No, and I am 90% sure of it.

(c)    Yes, and I am 100% sure of it.

(d)   No, and I am 100% sure of it.


 

  1. Construct a 90% confidence interval for the population mean, μ.  Assume the population has a normal distribution.  In a recent study of 22 eighth graders, the mean number of hours per week that they watched television was 19.6 with a standard deviation of 5.8 hours.

(a)    5.87 < μ < 7.98        (b)  18.63 < μ < 20.89       (c)  17.47 < μ < 21.73       (d) 19.62 < μ < 23.12

 

  1. A survey of 250 households showed 19 owned at least one gun.  Find a point estimate for p, the population proportion of households that own at least one gun.

(a)  0.076                           (b)  0.924                           (c)  0.082                           (d) 0.071

 

  1. A pollster wishes to estimate the proportion of United States voters who favor capital punishment.  How large a sample is needed in order to be 90% confident that the sample proportion will not differ from the true proportion by more than 2%?

(a)    3383                            (b)  1024                            (c)  21                                (d) 1692

 

  1. A manufacturer of golf equipment wishes to estimate the number of left-handed golfers.  How large a sample is needed in order to be 99% confident that the sample proportion will not differ from the true proportion by more than 3%?  A previous study indicates that the proportion of left-handed golfers is 10%.

(a)    737                              (b)  543                              (c)  664                              (d) 20