Do all the following problems.
I. Choose the best answer for each multiple choice. Please use CAPITAL letters to indicate your answer and write neatly. (30 points)
1. ____ 4.____ 7. ____ 10.____ 13. ____
2. ____ 5.____ 8. ____ 11.____ 14. ____
3. ____ 6.____ 9. ____ 12.____ 15. ____
Whenever the population standard deviation is unknown and the population has a normal or near-normal distribution, which distribution is used in developing an interval estimation?
A. standard distribution
B. z distribution
C. alpha distribution
D. t distribution
As the sample size increases, the margin of error
A. increases
B. decreases
C. stays the same
D. increases or decreases depending on the size of the mean
What type of error occurs if you fail to reject H0 when, in fact, it is not true?
A. Type II
B. Type I
C. either Type I or Type II, depending on the level of significance
D. either Type I or Type II, depending on whether the test is one tail or two tail
In general, higher confidence levels provide
A. wider confidence intervals
B. narrower confidence intervals
C. a smaller standard error
D. unbiased estimates
In order to test the following hypotheses at an α level of significance
H0: µ ≤ 100
Ha: µ > 100
the null hypothesis will be rejected if the test statistic Z is
A. ≥ Z_{α}
B. ≤ Z_{α}
C. ≤ - Z_{α}
D. < 100
The level of significance
A. can be any positive value
B. can be any value
C. is (1 - confidence level)
D. can be any value between -1.96 to 1.96
Which of the following does not need to be known in order to compute the p-value?
A. knowledge of whether the test is one-tailed or two-tailed
B. the value of the test statistic
C. the level of significance
D. None of these alternatives is correct.
If a hypothesis is not rejected at the 5% level of significance, it
A. will also not be rejected at the 1% level
B. will always be rejected at the 1% level
C. will sometimes be rejected at the 1% level
D. None of these alternatives is correct.
A sample of 20 items from a population with an unknown σ is selected in order to develop an interval estimate of µ. Which of the following is not necessary?
A. We must assume the population has a normal distribution.
B. We must use a t distribution.
C. Sample standard deviation must be used to estimate σ.
D. The sample must have a normal distribution.
When the level of confidence decreases, the margin of error
A. stays the same
B. becomes smaller
C. becomes larger
D. becomes smaller or larger, depending on the sample size
As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution
A. |
becomes larger |
B. |
becomes smaller |
C. |
stays the same |
D. |
None of these alternatives is correct. |
If an interval estimate is said to be constructed at the 90% confidence level, the confidence coefficient would be
A. |
0.1 |
B. |
0.95 |
C. |
0.9 |
D. |
0.05 |
The probability of committing a Type I error when the null hypothesis is true is
A. |
the confidence level |
B. |
β |
C. |
greater than 1 |
D. |
the Level of Significance |
A student believes that the average grade on the final examination in statistics is at least 85. She plans on taking a sample to test her belief. The correct set of hypotheses is
A. |
H_{0}: µ < 85 H_{a}: µ _{≥} 85 |
B. |
H_{0}: µ _{≤} 85 H_{a}: µ > 85 |
C. |
H_{0}: µ _{≥} 85 H_{a}: µ < 85 |
D. |
H_{0}: µ > 85 H_{a}: µ _{≤} 85 |
For a one-tailed test (lower tail) at 93.7% confidence, Z =
A. |
-1.86 |
B. |
-1.53 |
C. |
-1.96 |
D. |
-1.645 |
II. Problem Solving
A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. The standard deviation of the population is known to be 5.6 days.
State the null and alternative hypotheses. (6 points)
Compute the test statistic. (9 points)
What is the critical value? Using the critical value approach, test to determine if their advertisement is legitimate. Let α = .05. (6 points)
A random sample of 49 children with working mothers showed that they were absent from school an average of 6 days per term with a standard deviation of 1.8 days.
Write down the equation you should use to construct the confidence interval for the average number of days absent per term for all the children. (6 points)
Determine a 98% confidence interval estimate for the average number of days absent per term for all the children. (8 points)
Determine a 95% confidence interval estimate for the average number of days absent per term for all the children. (8 points)
Discuss why 98% and 95% confidence intervals are different. (6 points)
Automobiles manufactured by the Efficiency Company have been averaging 42 miles per gallon of gasoline in highway driving. It is believed that its new automobiles average more than 42 miles per gallon. An independent testing service road-tested 36 of the automobiles. The sample showed an average of 42.8 miles per gallon with a standard deviation of 1.2 miles per gallon. With a 0.05 level of significance using the p-value approach, test to determine whether or not the new automobiles actually do average more than 42 miles per gallon.
State the null and alternative hypotheses. (6 points)
Compute the test statistic. (9 points)
What is the p- value associated with the sample results? What is your conclusion based on the p-value? (6 points)
1. Whenever the population standard deviation is unknown and the population has a normal or near-normal distribution,
t distribution is used in developing an interval estimation .
2. As the sample size increases, the margin of error decreases.
3. fail to reject H0 when, in fact, it is not true - Type II error.
4. higher confidence levels provide wider confidence intervals .
5. the null hypothesis will be rejected if the test statistic Z is ≥ Zα.
6.The level of significance is (1 - confidence level)
7 the level of significance does not need to be known in order to compute the p-value .
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