a.
Explain the Type I error and Type II errors in the context of this problem situation.
a.
Explanation of Solution
Given info:
In a certain region two different companies have applied to provide cable television service. Suppose p be the proportion of all potential subscribers who favor the first company over the second. The null hypothesis is
Justification:
Type I error:
Reject the null hypothesis when it is actually true.
Type II error:
Fail to reject the null hypothesis when it is actually false.
Hypotheses:
Null hypothesis:
Alternative hypothesis:
If the hypothesis concluded that the proportion of all potential subscribers who favor the first company over the second differs from 50-50 proportion, but in reality, the proportion of all potential subscribers who favor the first company over the second is 50-50, then Type I error will arise.
If the hypothesis concluded that the proportion of all potential subscribers who favor the first company over the second is 50-50 proportion, but in reality, the proportion of all potential subscribers who favor the first company over the second differs from 50-50, then Type II error will arise.
b.
Find the values of X which will be at least as contradictory to
b.
Answer to Problem 11E
The all possible values which are at least contradictory as x = 6 are
Explanation of Solution
Calculation:
Assume X be the number in the sample who favor the first company. The samples are randomly selected with two possible outcomes.
Hence, X follows binomial distribution,
It is known that the mean of binomial random variable is
If the null hypothesis
Here, the values which are less than 6 are also as contradictory to reject the null hypothesis
The probability mass
The distance of 6 from 12.5 is
Hence, it can be said that the right side are also will be equally contradictory for 19 or more values.
Hence, the all possible values which are at least contradictory as x = 6 are
c.
Find the probability distribution of the test statistic when
Find the P-value for x = 6.
c.
Answer to Problem 11E
The distribution of X is,
The P-value for x = 6 is 0.014.
Explanation of Solution
Calculation:
From part b,
The null hypothesis is defined as
Thus, when the null hypothesis is true, then
Hence, when
That is,
P-value:
The probability of getting the value of the statistic that is as extreme as the observed statistic when the null hypothesis is true is called as p-value. Therefore, it assumes “null hypothesis is true”.
The P-value is,
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 6, 18
- Then, obtain the table value corresponding to p = 0.5.
The value of
Hence,
The P-value for x = 6 is 0.014.
d.
Find the probability of Type II error when p = 0.4, 0.3, 0.6 and 0.7.
d.
Answer to Problem 11E
The probability of Type II error for = 0.4, 0.3, 0.6, 0.7 are 0.846, 0.488, 0.846 and 0.488 respectively.
Explanation of Solution
Given info:
Calculation:
The P-value for two-tailed test is divided into two sides of the hypothesis test.
Thus, for two tailed test
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose p = 0.5.
- Locate the probability 0.022.
- The value of x is 7.
The distance of 7 from 12.5 is
Hence,
Type II error:
Fail to reject the null hypothesis when it is actually false.
The rejection rule of null hypothesis is
Hence for fail to reject the null hypothesis the range of x will be
Type II error for p:
Where,
Type II error for p = 0.4:
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 17, 7
- Then, obtain the table value corresponding to p = 0.4.
The value of
Hence,
Type II error for p = 0.3:
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 17, 7
- Then, obtain the table value corresponding to p = 0.3.
The value of
Hence,
Type II error for p = 0.6:
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 17, 7
- Then, obtain the table value corresponding to p = 0.6.
The value of
Hence,
Type II error for p = 0.7:
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 17, 7
- Then, obtain the table value corresponding to p = 0.7.
The value of
Hence,
Hence, the probability of Type II error for = 0.4, 0.3, 0.6, 0.7 are 0.846, 0.488, 0.846 and 0.488 respectively.
e.
Find the conclusion if 6 of the 25 queried favored company 1.
e.
Answer to Problem 11E
The proportion of all potential subscribers who favor the first company over the second is not 50-50 proportion.
Explanation of Solution
Calculation:
If 6 of the 25 queried favored company 1, then the P-value is
From part c,
Decision rule:
If
If
Conclusion:
Here, the P-value is less than 0.044
That is,
By rejection rule, “reject null hypothesis”.
Hence, the null hypothesis will be rejected.
Hence, the proportion of all potential subscribers who favor the first company over the second is not 50-50 proportion.
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Chapter 8 Solutions
Probability and Statistics for Engineering and the Sciences
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill