MTH 305 | Midterm Exam | Complete Solution

                                                                   Midterm Exam

                                                             (100 Total Points Possible )

Each question is worth 10 points. Include complete answers or calculations in the space below each question.

1. What is hypothesis testing? Explain in words the meaning and use a business example.

Hypothesis testing is the use of statistics to determine the probability that a given hypothesis is true.  Example:  ABC Corporation is a company that is focused on a stable workforce that has very little turnover. ABC has been in business for 50 years and has more than 10,000 employees. The company has always promoted the idea that its employees stay with them for a very long time, and it has used the following in its recruitment brochures. “The average tenure of our employees is 20 years”. Since ABC isn’t quite sure if that statement is still true, a random sample of 100 employees is taken and the average age turns out to be 19 years with a standard deviation of 2 years. Can ABC continue to make its claim, or does it need to make a change?

1. State the hypotheses:    Ho: u: =20 years                   H1:u: ≠ 20 years
2. Determine the test statistic. Since we are testing a population mean that is normally distributed, the appropriate test statistics:
3. Specify the significance level. Since the firm would like to keep its present message to new recruits, it select a fairly weak significance level (α=.05). Since this is a two-tailed test, half of the alpha will be assigned to each tail of the distribution. In this situation the critical values of Z = +1.96 and -1.96.
4. State the decision rule. If the computed value of Z  is greater than or equal to +1.96 or less than or equal to - 1.96, the null hypothesis is rejected
5. Reject or fail to reject the null. Since 2.5 is greater than 1.96, the null is rejected. The mean tenure is not 20 years, therefore ABC needs to change its statement.

1.  What is the definition of null hypothesis? 

Null hypothesis is a type of hypothesis used in statistics that proposes that no statistical significance exists in a set of a given observation. A proposition that undergoes verification to determine if it should be accepted or rejected in favor of an alternative proposition.

1.  What is the definition of the non-rejection region in hypothesis testing?

Non- rejection region: The set of values not in the rejection that leads to non- rejection of Ho.

1. The amount of time a financial consultant spends with each customer has a population standard deviation =25 minutes. You select a random sample of 100 customers and calculate the sample mean of time spent with a customer to be Xbar=60 minutes. At a 95% confidence level you calculate the confidence interval for the population mean u to be 55.1< u< 64.9.
2. Would you reject the null hypothesis that u= 70 at a 95% confidence level? Why?
3. What is the alternative hypothesis in part (a) above?
4. What is the formula of the test statistic for testing the hypothesis in part (a)?
5. What would happen to the confidence interval above if you take a larger sample of n= 225 customers?

1. Information given:

• A government inspector wants to know whether the amount of soft drink in a bottle at a soft drink plant averages 2 liters.
• Inspector takes random sample of 100 bottles (n=100) and gets sample mean of Xbar=1.99 liters and S=0.05 liters.

Explain how the inspector can answer his concern. You do not need to calculate anything, just show the complete steps for testing his problem using the standard hypothesis testing procedure. Show complete formulas.

1. In a problem # 5 above, suppose that the inspector gives you the results from Excel as shown below. What would you conclude about his main test given the results below? Why?

                                            Amount of soft drinks in bottle (in liters)

                                     Mean                                                            1.99

                                    Standard Error                                            0.005

                                    Median                                                          1.95

                                    Mode                                                              1.98

                                    Standard deviation                                       0.05

                                    Sample Variance                                       0.0025

                                     Kurtosis                                           1.102636883

                                    Skewness                                          1.301596705

                                   Range                                                                0.29

                                  Minimum                                                          1.87

                                  Maximum                                                          2.16

                                  Sum                                                                      199

                                 Count                                                                     100

                                  Confidence Level (95.0%)                 0.009921085

1. Suppose you were given the dataset below. Explain with the use of standard hypothesis testing and with statistics formulas how you would test the claim that “females do not spend the same amount of time checking email as males”. Do not explain it using Excel functions; rather show all the steps that you would take in hypothesis testing as shown in the chapters from the textbook. You do not need to compute anything. Explain any assumptions you needed to consider if that was the case.

Student ID            Gender         Hrs. checking E-mail/week     Age

      1                           F                                       5                             18

      2                           F                                       8                             19

      3                           F                                       14                           20

      4                           F                                       5                             23

      5                           F                                       10                           21

     6                            F                                        20                          22

     7                            F                                        14                          21

     8                            F                                         7                           18

     9                           F                                          7                            19

     10                         F                                          7                            20

     11                         F                                        10                            18

     12                         F                                          7                            20

     13                         F                                          8                            22

     14                         F                                          15                          19

      15                        F                                           10                          18

      16                       M                                           7                           20

      17                        M                                          15                         20

      18                        M                                          10                          20

      19                        M                                           7                            19

      20                         M                                           7                            19

       21                        M                                          15                          18 

                                                                                                                                                               6  

       22                               M                                       5                               20         

1. Suppose that the results from Excel for the problem in question #7 above is shown below in the table.

1. What would you conclude: do females spend the same amount checking emails as male? Why?
2. What do the Excel results mean that it was a t- test of two sample assuming unequal variances.

t-Test: Two Sample Assuming Unequal Variances

                                                    FEMALES               MALES

Mean                                 10.14285714             9.833333333

Variance                            17.36263736            18.56666667

Observations                                       14                                6

Hypothesized Mean Difference          0

df                                                             9

t Stat                                      0.148668139

P(T<=t) one tail                    0.442546652

t Critical one tail                    1.83112923

P(T<=t) two-tail                   0.885093304

t Critical two-tail                2.262157158

1. Using the Excel results shown in problem#8 above, what can we do to check whether female spending time on emails varies the same as male spending time? Explain in detail using steps in hypothesis testing.                                                                                                                                            
2. What would happen to your test statistics from problem #9 above if variance of females was doubled the amount shown in problem#8? Why?