Math 115 Practice Test 1

1. A marketing researcher surveys 350 consumers about their shopping habits, recording each person’s household income (in dollars), number of online purchases per month, preferred payment method (credit card, debit card, cash, digital wallet), and whether they use a shopping rewards program (yes/no).

(a) What are the cases in this study? Circle one.

       i.          The 4 variables measured

     ii.          The 350 consumers surveyed

    iii.          All consumers in the population

    iv.          The marketing researcher

(b) Identify each variable as categorical or numerical. Circle one for each.

2. Classify each variable as nominal categorical, ordinal categorical, discrete numerical, or continuous numerical. Circle one for each.

a.     Eye color (blue, brown, green, hazel)

                           i.          Nominal categorical

                         ii.          Ordinal categorical

                        iii.          Discrete numerical

                        iv.          Continuous numerical

b.     Number of pets owned

                           i.          Nominal categorical

                         ii.          Ordinal categorical

                        iii.          Discrete numerical

                        iv.          Continuous numerical

c.     Distance traveled to work (in miles)

                           i.          Nominal categorical

                         ii.          Ordinal categorical

                        iii.          Discrete numerical

                        iv.          Continuous numerical

d.     Customer satisfaction rating (very dissatisfied, dissatisfied, neutral, satisfied, very satisfied)

                           i.          Nominal categorical

                         ii.          Ordinal categorical

                        iii.          Discrete numerical

                        iv.          Continuous numerical

3. A survey of 300 employees at a large company asked about their primary mode of transportation to work. The results are shown below:

Transportation Mode

Frequency

Bicycle

46

Car

133

Public Transit

91

Walking

30

Total

300

 

(a) What proportion of employees primarily use public transit? (Round to three decimal places.)

Answer: _________

(b) How many employees primarily bicycle to work?

Answer: _________

(c) What percentage of employees primarily drive a car? (Round to one decimal place.)

Answer: _________%

4. A meteorologist records the following 10 daily high temperatures (in degrees Fahrenheit) for a city: 58, 62, 64, 65, 66, 67, 68, 69, 70, 95.

(a) Calculate the mean. (Round to one decimal place.)

Answer: _________

(b) Calculate the median.

Answer: _________

(c) Which measure (mean or median) is larger? Circle one.

       i.          Mean

     ii.          Median

    iii.          They are equal

(d) Based on your answer to (c), is this distribution left-skewed, right-skewed, or symmetric? Circle one.

       i.          Left-skewed

     ii.          Right-skewed

    iii.          Symmetric

https://math.hope.edu/yurk/115/practiceTests/practice_test1_files/figure-html/unnamed-chunk-5-1.png

 

5. Based on the boxplot above:

(a) What is the approximate median commute time? (Round to one decimal place.)

Answer: _________

(b) What is the approximate IQR? (Round to one decimal place.)

Answer: _________

6. A researcher conducts a hypothesis test and obtains a p-value of 0.72. Which statement best describes this result? Circle one.

A.    There is strong evidence against the null hypothesis

B.     There is moderate evidence against the null hypothesis

C.    There is weak evidence against the null hypothesis

D.    There is essentially no evidence against the null hypothesis

7. A researcher rejects the null hypothesis at the α = 0.05 significance level. Which of the following is a correct interpretation? Circle one.

A.    The null hypothesis has been proven false

B.     The alternative hypothesis has been proven true

C.    The data provide convincing evidence against the null hypothesis

D.    The probability that the null hypothesis is true is less than 0.05

8. A 99% confidence interval is constructed for a population mean. Compared to a 95% confidence interval using the same data, the 99% interval would be: Circle one.

A.    Narrower because we need more precision for higher confidence

B.     Wider because we need a larger range to be more confident

C.    The same width because both use the same data

D.    Cannot be determined without knowing the sample size

9. A 95% confidence interval for a population mean is (42, 58). Which of the following statements is TRUE? Circle all that apply.

A.    The sample mean is 50

B.     There is a 95% probability that the true mean is between 42 and 58

C.    We are 95% confident that the true mean is between 42 and 58

D.    95% of all sample means fall between 42 and 58

10. A movie theater surveys 180 patrons and finds that 99 prefer watching movies in 3D. A bootstrap 95% confidence interval for the true proportion who prefer 3D is (0.48, 0.62).

Based on this confidence interval, at the significance level α = 0.05 would you reject H0 : p = 0.5 in favor of HA : p ≠ 0.5  ? Circle one.

A.    Yes, reject H0 because 0.5 is not in the confidence interval

B.     Yes, reject H0 because 0.5 is in the confidence interval

C.    No, fail to reject H0 because 0.5 is not in the confidence interval

D.    No, fail to reject H0 because 0.5 is in the confidence interval

11. Which of the following would make a confidence interval WIDER? Circle all that apply.

A.    Increasing the sample size

B.     Decreasing the sample size

C.    Increasing the confidence level from 95% to 99%

D.    Decreasing the confidence level from 95% to 90%

12. A researcher reports: “The average GPA of students at State University is 3.2.” This value is a: Circle one.

A.    Parameter, because it describes the entire population of State University students

B.     Statistic, because it was calculated from a sample of students

C.    Parameter if it was calculated from all students, or a statistic if from a sample

D.    Neither a parameter nor a statistic

13. Determine whether each statement about confidence intervals is TRUE or FALSE. Circle one for each.

(a) Increasing the sample size will make a confidence interval narrower.

True / False

(b) A 99% confidence interval is narrower than a 95% confidence interval.

True / False

(c) If a 95% CI for a proportion is (0.52, 0.68), then 0.50 is a plausible value for the population proportion. True / False

14. A restaurant chain claims that 70% of customers are satisfied with their service. A regional manager suspects the true satisfaction rate in her region is lower than 70%. She surveys a random sample of 120 customers and finds that 75 report being satisfied.

Below is a null distribution showing 1000 simulated proportions under the assumption that the true satisfaction rate is 70%.

https://math.hope.edu/yurk/115/practiceTests/practice_test1_files/figure-html/unnamed-chunk-10-1.png

Use this histogram and the information above to answer the following questions.

 

(a) What is the parameter of interest in this study? Circle one.

       i.          The proportion of the 120 sampled customers who reported being satisfied

     ii.          The true proportion of all customers in this region who are satisfied with service

    iii.          The 70% satisfaction rate claimed by the restaurant chain

    iv.          The number of satisfied customers in the sample

(b) Which pair of hypotheses is appropriate for this research question? Circle one.

(c) Calculate the sample proportion  of customers in the sample who reported being satisfied. (Round to two decimal places.)

Answer: _________

(d) Using the histogram, estimate the p-value for this hypothesis test. (Round to three decimal places.)

Answer: _________

(e) Which of the following correctly interprets the p-value in the context of this problem? Circle one.

       i.          The probability that the true satisfaction rate is 70%

     ii.          The probability that the true satisfaction rate is less than 70%

    iii.          The probability of observing a sample proportion of  or lower, assuming the true satisfaction rate is 70%

    iv.          The probability of observing a sample proportion of exactly , assuming the true satisfaction rate is less than 70%

(f) Using a significance level of α = 0.05, what is the appropriate decision for this hypothesis test? Circle one.

       i.          Reject H0 

     ii.          Fail to reject H0 

    iii.          Accept H0 

(g) Based on your decision in part (f), which conclusion is appropriate? Circle one.

       i.          There is convincing evidence that the true satisfaction rate is less than 70%.

     ii.          There is not enough evidence to conclude that the true satisfaction rate is less than 70%.

    iii.          We have proved that the true satisfaction rate is exactly 70%.

    iv.          There is convincing evidence that the true satisfaction rate is greater than 70%.

15. A dentist wants to estimate the proportion of patients who floss daily. She surveys a random sample of 80 patients and finds that 52 report flossing daily.

Below is a bootstrap distribution showing 1000 bootstrap proportions based on resampling from the original sample with replacement.

https://math.hope.edu/yurk/115/practiceTests/practice_test1_files/figure-html/unnamed-chunk-12-1.png

Use this histogram and the information above to answer the following questions.

(a) What is the parameter of interest in this study? Circle one.

       i.          The proportion of the 80 sampled patients who report flossing daily

     ii.          The 52 patients in the sample who floss daily

    iii.          The true proportion of all patients who floss daily

    iv.          The average number of times patients floss per week

(b) Calculate the sample proportion of patients in the sample who report flossing daily. (Round to two decimal places.)

Answer: _________

(c) To construct a 95% confidence interval, which percentiles of the bootstrap distribution should be used? Circle one.

       i.          5th and 95th percentiles

     ii.          2.5th and 97.5th percentiles

    iii.          0.5th and 99.5th percentiles

    iv.          25th and 75th percentiles

(d) Using the histogram, estimate the lower bound (2.5th percentile) of the 95% confidence interval. (Round to two decimal places.)

Answer: _________

(e) Using the histogram, estimate the upper bound (97.5th percentile) of the 95% confidence interval. (Round to two decimal places.)

Answer: _________

 

 

 

(f) Which of the following is the correct interpretation of this 95% confidence interval? Circle one.

       i.          There is a 95% probability that the true proportion of patients who floss daily is in this interval.

     ii.          95% of all patients floss daily.

    iii.          We are 95% confident that the true proportion of all patients who floss daily is in this interval.

    iv.          95% of samples would have a proportion of patients who floss daily between these two values.

(g) Based on your confidence interval, is 0.60 (60%) a plausible value for the true proportion of patients who floss daily? Circle one.

       i.          Yes, because 0.60 is inside the confidence interval

     ii.          No, because 0.60 is outside the confidence interval

    iii.          Yes, because 0.60 is close to our sample proportion

    iv.          No, because we cannot make claims about specific values from a confidence interval