Explain what Type I error means in the context of this problem.


Test of μ = 50 vs ≠ 50

The assumed standard deviation = 10

 

 

Variable   N   Mean  StDev  SE Mean      95% CI          Z      P

C1        16  47.21  11.67     2.50  (42.31, 52.11)  -1.12  0.264

C2        16  51.49  10.92     2.50  (46.59, 56.39)   0.59  0.552

C3        16  51.15  13.13     2.50  (46.25, 56.05)   0.46  0.646

C4        16  49.82   8.29     2.50  (44.92, 54.72)  -0.07  0.942

C5        16  52.19   9.63     2.50  (47.29, 57.09)   0.88  0.381

C6        16  46.03   8.69     2.50  (41.13, 50.93)  -1.59  0.112

C7        16  45.10   9.84     2.50  (40.20, 50.00)  -1.96  0.050

C8        16  50.79  10.69     2.50  (45.89, 55.69)   0.32  0.751

C9        16  51.48   8.93     2.50  (46.58, 56.38)   0.59  0.554

C10       16  53.09   7.37     2.50  (48.19, 57.99)   1.23  0.217

C11       16  50.13   8.12     2.50  (45.23, 55.03)   0.05  0.960

C12       16  50.08   9.33     2.50  (45.18, 54.98)   0.03  0.974

C13       16  50.96   9.31     2.50  (46.06, 55.86)   0.38  0.701

C14       16  53.57  12.94     2.50  (48.67, 58.47)   1.43  0.153

C15       16  50.71  11.22     2.50  (45.81, 55.61)   0.29  0.775

C16       16  49.10   9.04     2.50  (44.20, 54.00)  -0.36  0.719

C17       16  51.37   8.67     2.50  (46.47, 56.27)   0.55  0.584

C18       16  48.85   9.25     2.50  (43.95, 53.75)  -0.46  0.645

C19       16  48.81   9.80     2.50  (43.91, 53.71)  -0.48  0.634

C20       16  48.42   5.25     2.50  (43.52, 53.32)  -0.63  0.528

C21       16  49.96  11.65     2.50  (45.06, 54.86)  -0.01  0.989

C22       16  51.06  11.79     2.50  (46.16, 55.96)   0.43  0.670

C23       16  55.63   9.12     2.50  (50.73, 60.53)   2.25  0.024

C24       16  51.01   8.65     2.50  (46.11, 55.91)   0.40  0.687

C25       16  50.01   9.96     2.50  (45.11, 54.91)   0.00  0.996

C26       16  51.13   8.77     2.50  (46.23, 56.03)   0.45  0.651

C27       16  53.32  10.58     2.50  (48.42, 58.22)   1.33  0.184

C28       16  48.86  10.71     2.50  (43.96, 53.76)  -0.46  0.649

C29       16  46.95   7.84     2.50  (42.05, 51.85)  -1.22  0.223

C30       16  51.81   8.49     2.50  (46.91, 56.71)   0.72  0.470

 

Now based on the values of the 30 test statistics, answer these questions.

(A) How many researchers would reject H0. That is, how many of them made an “incorrect decision”? ……..

(B) If we change the level of the test from α = 0.05 to α = 0.001 then, does this change any of your decisions to reject or not reject H0 ? ………..
(C) In general, should the number of rejections increase or de-crease if α = 0.001 is used instead of α = 0.05? ………….

 

 

 

 

Results for: Worksheet 2

 

One-Sample Z: C1, C2, C3, C4, C5, C6, C7, C8, …

 

Test of μ = 50 vs ≠ 50

The assumed standard deviation = 10

 

 

Variable   N   Mean  StDev  SE Mean      95% CI          Z      P

C1        16  53.04  11.53     2.50  (48.14, 57.94)   1.22  0.224

C2        16  50.53  12.80     2.50  (45.63, 55.43)   0.21  0.832

C3        16  52.28  14.36     2.50  (47.38, 57.18)   0.91  0.362

C4        16  50.85   7.81     2.50  (45.95, 55.75)   0.34  0.734

C5        16  50.24  11.92     2.50  (45.34, 55.14)   0.10  0.923

C6        16  48.41  10.56     2.50  (43.51, 53.31)  -0.64  0.525

C7        16  51.80  11.90     2.50  (46.90, 56.70)   0.72  0.471

C8        16  57.36   9.19     2.50  (52.46, 62.26)   2.94  0.003

C9        16  47.54  10.93     2.50  (42.64, 52.44)  -0.98  0.325

C10       16  54.86   9.12     2.50  (49.96, 59.76)   1.94  0.052

C11       16  54.77   9.27     2.50  (49.87, 59.67)   1.91  0.057

C12       16  50.39  11.51     2.50  (45.49, 55.29)   0.16  0.876

C13       16  50.67   7.94     2.50  (45.77, 55.57)   0.27  0.789

C14       16  55.83  10.80     2.50  (50.93, 60.73)   2.33  0.020

C15       16  54.01  10.05     2.50  (49.11, 58.91)   1.61  0.108

C16       16  55.27  12.24     2.50  (50.37, 60.16)   2.11  0.035

C17       16  49.03  12.04     2.50  (44.13, 53.93)  -0.39  0.698

C18       16  50.78   9.75     2.50  (45.88, 55.68)   0.31  0.756

C19       16  52.70   8.58     2.50  (47.80, 57.60)   1.08  0.281

C20       16  53.36   9.61     2.50  (48.46, 58.26)   1.35  0.179

C21       16  49.12   7.53     2.50  (44.22, 54.02)  -0.35  0.726

C22       16  53.62   9.01     2.50  (48.72, 58.52)   1.45  0.147

C23       16  52.85   9.69     2.50  (47.95, 57.75)   1.14  0.255

C24       16  48.39  13.29     2.50  (43.49, 53.29)  -0.64  0.521

C25       16  52.49   9.25     2.50  (47.59, 57.39)   0.99  0.320

C26       16  49.86  12.00     2.50  (44.96, 54.76)  -0.06  0.956

C27       16  51.55  11.96     2.50  (46.65, 56.45)   0.62  0.536

C28       16  54.48   7.26     2.50  (49.58, 59.38)   1.79  0.073

C29       16  51.82   8.13     2.50  (46.92, 56.72)   0.73  0.468

C30       16  51.89  11.28     2.50  (46.99, 56.79)   0.76  0.449

 

 

(D) If after a while we realized that the actual mean of the population is currently μ = 52 dollars per hour and it is no longer 50. Once again, using α = 0.05 and assuming σ is still 10 dollars per hour, in how many tests did you reject H0? ………….

(E) A rejection of H0 in part (A) is a “correct decision”. True or False? ………
(F) A rejection of H0 in part (D) is a “correct decision”. True or False? ……….

 

 

 

 

One-Sample T: C1, C2, C3, C4, C5, C6, C7, C8, …

 

Test of μ = 50 vs ≠ 50

 

 

Variable   N   Mean  StDev  SE Mean      95% CI          T      P

C1        16  49.78   8.31     2.08  (45.35, 54.20)  -0.11  0.916

C2        16  48.78   8.36     2.09  (44.33, 53.24)  -0.58  0.569

C3        16  46.76   9.94     2.48  (41.46, 52.05)  -1.30  0.212

C4        16  48.45  10.02     2.51  (43.11, 53.79)  -0.62  0.547

C5        16  49.66   7.42     1.86  (45.70, 53.62)  -0.18  0.857

C6        16  48.97   7.95     1.99  (44.73, 53.20)  -0.52  0.611

C7        16  47.90  10.00     2.50  (42.57, 53.23)  -0.84  0.413

C8        16  47.31   9.69     2.42  (42.15, 52.48)  -1.11  0.285

C9        16  49.30  12.22     3.06  (42.79, 55.81)  -0.23  0.822

C10       16  47.73   9.62     2.41  (42.60, 52.86)  -0.94  0.360

C11       16  52.47  11.26     2.82  (46.47, 58.47)   0.88  0.394

C12       16  44.25  13.11     3.28  (37.26, 51.23)  -1.75  0.100

C13       16  47.21  10.86     2.72  (41.43, 53.00)  -1.03  0.321

C14       16  48.91  10.92     2.73  (43.09, 54.73)  -0.40  0.695

C15       16  47.33  12.47     3.12  (40.68, 53.97)  -0.86  0.405

C16       16  47.83   8.09     2.02  (43.52, 52.14)  -1.08  0.299

C17       16  51.65   8.38     2.09  (47.19, 56.12)   0.79  0.442

C18       16  49.15   7.86     1.97  (44.96, 53.35)  -0.43  0.673

C19       16  51.86   9.92     2.48  (46.57, 57.14)   0.75  0.466

C20       16  47.20  10.72     2.68  (41.49, 52.91)  -1.04  0.313

C21       16  46.87   8.91     2.23  (42.12, 51.62)  -1.40  0.180

C22       16  50.88   6.68     1.67  (47.32, 54.44)   0.53  0.606

C23       16  48.66  12.16     3.04  (42.18, 55.14)  -0.44  0.666

C24       16  55.54   8.67     2.17  (50.92, 60.17)   2.56  0.022

C25       16  51.28  12.36     3.09  (44.69, 57.87)   0.41  0.685

C26       16  52.08   7.44     1.86  (48.12, 56.04)   1.12  0.281

C27       16  49.84   6.21     1.55  (46.53, 53.15)  -0.10  0.919

C28       16  53.80   9.26     2.31  (48.86, 58.73)   1.64  0.122

C29       16  47.33  10.44     2.61  (41.77, 52.90)  -1.02  0.323

C30       16  52.64   8.92     2.23  (47.89, 57.39)   1.18  0.255

 

 

Repeat parts (A), (B), and (C) of Question 1, using ttest in- stead of ztest, and answer (Thus ‘ztest 50 10 c1-c30’ changes to ‘ttest 50 c1-c30’)
(A) In how many tests did you reject H0. That is, how many times did you make an “incorrect decision”? ……….
(B) Suppose you used α = 0.00008 instead of α = 0.05. Does this change any of your decisions to reject or not? ……….. (C) In general, should the number of rejections increase or decrease if α = 0.00008 is used instead of α = 0.05? …………

 

 

 

 

 

  1. I am deciding on whether to invest 400,000 CAD to open a convenience store in particular spot in Ottawa. I know that the business will be profitable, with income of 70,000 CAD per year, if the store will have in average μ = 60 or more customers per day. If I am convinced that the business will be profitable then I will go ahead and open the store otherwise I won’t. Hence I am dealing with this hypothesis testing problem.

H0 : μ = 60 vs Ha : μ < 60

(A) Explain what Type I error means in the context of this problem.
(B) What is the consequence of Type I error.
(C) Explain what Type II error means in this context.

(D) What is the consequence of Type II error in the context of this problem.
(E) Which error is more expensive in your opinion. Explain.

2.A recent study conducted by the government attempts to determine proportion of people who support further increase in cigarette taxes. In this study, 2500 voting age citizens were sampled, and it was found that 1900 of them were in favor of an increase in cigarette taxes. 
At level α = 0.05, do you believe that 78% of all citizens are in favor of an increase in cigarette taxes.
(A) State the null and alternative hypotheses and compute the p-value. 
(B) What is the smallest α you need to reject the null hypothesis.

3.To study the average amount of debts in Canada, 100 Canadians were surveyed. The amount of debt of each one was recorded. The sample gave an average of 28110 CAD with standard deviation of 3500 CAD. It is believed that the actual mean value of the debt in Canada is 27500. Given our data, we would like to test if the actual mean of debt is higher than 27500. 
(A) State the null and alternative hypotheses and compute the value of the test statistic.
(B) Compute the p-value for this test
(C) Draw a conclusion by comparing the p-value you obtained in part (B) to α = 0.05.

4      -A caviar producing company packs its product in containers. They claim that the average weight of the caviar in the containers is 1 kilogram. They sell each container for 8,000 CAD. We had a budget of approximately 90,000 CAD. So, we could buy 11 of the containers and weighted the caviar in each. Here are the results in Kilograms: 
0.89, 1.01, 1.00, 0.90, 0.90, 0.91, 0.91, 0.89, 0.95, 1.00, 0.96. 
If we assume that the wights are normally distributed is there sufficient evidence for us to believe that the actual average weight of caviar in the containers is less that what the producer claims it is. Use the critical value approach with α = 0.05

  1. The mid-distance running coach, Zdravko Popovich, for the Olympic team of an eastern European country claims that his six-month training program significantly reduces the average time to complete a 1500-meter run. Four mid-distance runners were randomly selected before they were trained with coach Popovich’s six-month training program and their completion time of 1500-meter run was recorded (in minutes). After six months of training under coach Popovich, the same four runners’ 1500 meter run time was recorded again. The results are given below.

 

 

 

Runner 1 2 3 4
Completion time before training

Completion time after training

6.0

5.5

7.5

7.1

6.2

6.2

6.8

6.4

 

Assume that the times before training and after training are normally distributed.

(A) Are the two samples independent? Explain.

(B) Construct 95% confidence interval for the difference between the actual mean of completion time before training and the actual mean of completion time after training.

 

(C) Based on the confidence interval in part (B) can we conclude that there is no significant difference between the between the actual mean of completion time before training and the actual mean of completion time after training. Hint: Check and see if your confidence interval in part (B) includes 0.

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