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# Type I and II errors and Confidence interval

1. You have been asked to evaluate if a particular coin is fair or not. You have decided to use the following test:
Accept that the coin is fair if in 30 tosses the coin gives between 11 and 19 heads (inclusive); reject the hypothesis of fairness otherwise.

Compute the Type I error rate of this test. Interpret, in plain words, what the computed number would mean.

Compute the power (Type II error) of this test at p=.3, .49, .51, and .7. Interpret what the computed numbers mean. In addition, comment on any interesting patterns you found in these numbers. Then explain what is the cause of the patterns you discovered.

2. The time till first failure of a brand of HD plasma tvs is advertised to be more than 8 years. Suppose you sample 15 tvs of this type, keep track of them, and accept the advertised claim if the mean time to the first failure is larger than 8.5 years.
Plot the type I error probability and the power function of this particular test.

3. A t-statistic for testing H0: &#956;=3.5 is to be computed based on 20 observations with a mean of 4 and a standard deviation of 1.
Then, a 21st observation was collected and this new observation was 10. Recompute the t-statistic when all 21 observations are used.

Comment on what you find.

4. In 8 hospital rooms that are carpeted, and another 8 that are uncarpeted, bacteria per cubic ft. were counted, and the data are as follows:

Carpeted rooms: 11.8, 8.2, 7.1, 13, 10.8, 10.1, 14.6, 14
Uncarpeted rooms: 12.1, 8.3, 3.8, 7.2, 12, 11.1, 10.1, 13.7
Compute a nominal 90% confidence interval for the difference in mean bacterial count per cubic ft. in carpeted and uncarpeted hospital rooms. Discuss the pros and cons of each type of confidence interval for this particular application. Discuss what assumptions you would be concerned about and whether you can test them, if you have any doubts.

#### Solution Preview

You have been asked to evaluate if a particular coin is fair or not. You have decided to use the following test:
Accept that the coin is fair if in 30 tosses the coin gives between 11 and 19 heads (inclusive); reject the hypothesis of fairness otherwise.
Compute the Type I error rate of this test. Interpret, in plain words, what the computed number would mean.
The probability density function of binomial random variable with parameter n and p is given by the formula. .
Under H0. P =0.5

X P(X)
0 0.00000
1 0.00000
2 0.00000
3 0.00000
4 0.00003
5 0.00013
6 0.00055
7 0.00190
8 0.00545
9 0.01332
10 0.02798
11 0.05088
12 0.08055
13 0.11154
14 0.13544
15 0.14446
16 0.13544
17 0.11154
18 0.08055
19 0.05088
20 0.02798
21 0.01332
22 0.00545
23 0.00190
24 0.00055
25 0.00013
26 0.00003
27 0.00000
28 0.00000
29 0.00000
30 0.00000

Thus probability of rejecting H0 under p =0.5 = 1-0.9013 =0.0987. Thus there is a chance of 9.87% in concluding that the coin is not fair when in fact the coin is a fair one.
Compute the power (Type II error) of this test at p=.3, .49, .51, and .7. Interpret what the computed numbers mean. In addition, comment on any interesting patterns you found in these numbers. Then explain what is the cause of the patterns you discovered.
Type II error occur when concluding that the coin is fair when in fact the coin is biased. The probability for the accepting that for different values of p are given ...

#### Solution Summary

Step by step method for computing test statistic for hypothesis test is given in the answer.

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