STAT217 Tutorial sheet #1 Solutions

 

1)      P(X < 35) = 0.6554.

2)      P(X > 50) =  1 – P(X < 50) = 1 – 0.9452 = 0.0548.

3)      P(X < 25 < X < 50) = P(X < 50) – P(X < 25). We already know that P(X > 50) = 0.0548 from the previous question. So, P(X < 50) = 0.9452. 

P(X < 25) = 0.3446.

Finally, P(X < 50) – P(X < 25) = 0.9452 – 0.3446 = 0.6006.

4)      Since we’re interesting in the top 10% of sales, we need the 90th percentile. Using Excel, we find Z = 1.2816. So X = 30 + 1.2816(12.5) = 46.02 which rounds to 46.

5)      Using the central limit theorem, Z = (25 – 30)/(12.5/sqrt(50)) = -2.8284. P(Xbar < 25) = P(Z < -2.8284) = 0.0023.

6)      Lower limit = 29.50 – 1.96(12.5)/sqrt(50) = 29.50 – 3.46 = 26.04

Upper limit = 29.50 + 3.46 = 32.96.

With 95% confidence, the average sale at this stores ranges from $26.04 to $32.96.

7)      If the level of confidence is increased to 99%, the Z value changes from 1.96 to 2.5758. The margin of error then becomes 2.5758(12.5)/sqrt(50) = 4.55. The difference between the new margin of error and the old one is 4.55 – 3.46 = 1.09.

8)      Sample size = n = (1.96(12.50)/2.5)2 = 9.82 = 96.04 which we round up to 97.

9)      The thing to keep in mind is that the Central Limit theorem works regardless of the parent distribution if the sample size is large enough.

Z1 = (75.3 – 75)/(1.2/sqrt(60)) = 1.9365

Z2 = (74.7 – 75)/(1.2/sqrt(60)) = -1.9365

So P(74.7 < Xbar < 75.3) = P(-1.9365 < Z < 1.9365) = P(Z < 1.9365) – P(Z < -1.9365)

= 0.9736 – 0.0264 = 0.9472

10)  Lower limit = 0.42 – 1.96sqrt(0.42(0.58)/200) = 0.42 – 0.0684 = 0.3516

Upper limit = 0.42 + 0.0684 = 0.4884

With 95% confidence, the percentage of Ipod users in the 18-34 age bracket ranges from 35.16% to 48.84%.

11)  Sample size = n = (1.96/0.03)2(0.42)(0.58) = 1039.8 which we round to 1040.

12)  E = 1.96sqrt(0.42*0.58/500) = 0.0433 or 4.33%

13)  p* = (3 + 2)/(400 + 4) = 5/404

Lower limit = 5/404 – 1.96sqrt((5/404)(399/404)/404) = 0.0124 – 0.0108 = 0.0016

Upper limit = 0.0124 + 0.0108 = 0.0232

With 95% confidence, the percentage of Canadians who watch Space is between 0.16% and 2.32%.

14)  Sample size = n = (1.96/0.001)2(5/404)(399/404) - 4 = 46,952.1 which we round to 46,953.

15)  Solution 1: decrease level of confidence to 90%

Sample size = (1.6449/0.001)2(5/404)(399/404) - 4 = 33,067.9 which we round to 33,068.

Solution 2: keep confidence at 95% and increase margin of error to 0.5%

Sample size = (1.96/0.005)2(5/404)(399/404) - 4 = 1,874.2 which we round to 1,875.

Increasing the margin of error has the greater effect on decreasing the sample size.

16)  We have 1000 = (1.96/E)2(5/404)(399/404) from which we get E = 1.96sqrt((5/404)(399/404)/1000) = 0.0069 = 0.69%