Part A-
Q1- 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 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?
Q2 - 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?
Part B-
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
5. 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
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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.