# Point Estimator for Uniformly Distributed Interval

Question Details:

Let X1, X2, ...., Xn be uniformly distributed on the interval (0 to a). Recall that the maximum likelihood estimator of a is â =max(Xi).

a) Argue intuitively why â cannot be an unbiased estimator for a.

b) Suppose that E(â ) = na /(n+1).

Is it reasonable that â consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.

c) Propose an unbiased estimator for a.

d) Let Y= max(Xi). Use the fact that Y≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y. Then show that the probability density function of Y is: f(y) = nyn-1/ an for (0 ≤ y ≤ a) and f(y) = 0 otherwise. Use this result to show that the maximum likelihood estimator for a is biased.

e) We have two unbiased estimators for a: the moment estimator â1 = 2Χbar and

â2 = [(n+1)/n] max (Xi), where max(Xi) is the largest observation in a random

sample of size n. It can be shown that V(â1) = a2/3n and that V(â2) = a2/[n(n+2)].

Show that if n>1, â2 is a better estimator than â1. In what sense is it a better estimator of a?

https://brainmass.com/statistics/maximum-likelihood-estimation/point-estimator-192513

#### Solution Summary

The solution determines the point estimator for a uniformly distributed interval.

Point estimate/hypothesis testing...

1. Which of the following is not a characteristic of the normal probability distribution?

a. The mean, median, and the mode are equal

b. The mean of the distribution can be negative, zero, or positive

c. The distribution is symmetrical

d. The standard deviation must be 1

e. None of the above answers is correct.

2. Which of the following is not a characteristic of the normal probability distribution?

a. symmetry

b. The total area under the curve is always equal to 1.

c. 99.72% of the time the random variable assumes a value within plus or minus 1 standard deviation of its mean

d. The mean is equal to the median, which is also equal to the mode.

e. None of the above answers is correct.

3. The level of significance is the

a. Maximum allowable probability of Type II error

b. Maximum allowable probability of Type I error

c. Same as the confidence coefiicient

d. Same as the p-value

e. None of the above answers is correct.

4. If two independent large samples are taken from two populations, the sampling distribution of the difference between the two sample means

a. Can be approximated by a Poisson distribution

b. Will have a variance of one

c. Can be approximated by a normal distribution

d. Will have a mean of one

e. None of the above answers is correct.

5. To compute an interval estimate for the difference between the means of two populations when samples are small, the t distribution can be used if it can be assumed that

a. The populations are normally distributed

b. The variances are equal

c. Both a and b are satisfied

d. The population means are equal

e. None of the above answers is correct.

6. If we are interested in testing whether the mean of population 1 is smaller than the mean of population 2, the

a. Null hypothesis should state µ1 - µ2 < 0

b. Null hypothesis should state µ1 - µ2 > 0

c. Alternative hypothesis should state µ1 - µ2 ≥ 0

d. Alternative hypothesis should state µ1 - µ2 ≤ 0

e. None of the above answers is correct.

7. The pooled variance is appropriate whenever the two populations

a. Are normally distributed

b. Have equal variances

c. Meet both requirements stated in a and b

d. None of the above answers is correct.

8. Ten percent of all employees at a large corporation call in sick on Mondays. A sample of 144 employees' records is taken on a Monday. The probability that the number of employees calling in sick is greater than 22 is

a. 0.0174

b. 0.0244

c. 0.4756

d. 0.9756

e. None of the above answers is correct.

9. The travel time for a college student traveling between her home and her college is uniformly distributed between 40 and 90 minutes. The probability that she will finish her trip in 80 minutes or less is

a. 0.02

b. 0.8

c. 0.2

d. 1

e. None of the above answers is correct.

10. The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a standard deviation of 2 ounces. What is the probability that a randomly selected item will weigh more than 10 ounces?

a. 0.3413

b. 0.8413

c. 0.1587

d. 0.5000

e. None of the above answers is correct.

11. A population has a mean of 180 and a standard deviation of 24. A sample of 64 observations will be taken. The probability that the sample mean will be between 183 and 186 is

a. 0.1359

b. 0.8185

c. 0.3413

d. 0.4772

e. None of these alternatives is correct.

12. From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately

a. 1.1022

b. 2

c. 30

d. 1.4847

e. 0.45

13. A sample of 92 observations is taken from an infinite population. The sampling distribution of is approximately

a. normal because is always approximately normally distributed

b. normal because the sample size is small in comparison to the population size

c. normal because of the central limit theorem

d. None of these alternatives is correct.

14. A sample of 25 observations is taken from an infinite population. The sampling distribution of p

a. Not normal since n< 30

b. Approximately normal because p is always normally distributed

c. Approximately normal if np ≥ 5 and n(1-P) ≥ 5

d. Approximately normal if np > 30 and n(1-P) > 30

e. None of the above answers is correct.

15. A researcher is interested in determining the average number of years employees of a company stay with the company. If past information shows a standard deviation of 7 months, what size sample should be taken so that at 95% confidence the margin of error will be 2 months or less?

16. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The average age in the sample was 22.5 years with a standard deviation of 5 years. Construct an 80% confidence interval for the true average age of the consumers.

17. You are given the following information obtained from a random sample of 4 observations.

25 47 32 56

What is the point estimate of µ?

18. If the standard deviation of the lifetimes of vacuum cleaners is estimated to be 300 hours, how large of a sample must be taken in order to be 97% confident that the margin of error will not exceed 40 hours?

19. Consider the following hypothesis test:

H0: p = 0.5

Ha: p ≠ 0.5

A sample of 800 provided a sample proportion of 0.58.

Using α = 0.05, what is the rejection rule?

20. A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and 80 indicated they like the taste. Determine the p-value.

21. A student believes that the average grade on the statistics final examination is 87. A sample of 36 final examinations is taken. The average grade in the sample is 83.96 with a standard deviation of 12. Compute the probability of a Type II error if the average grade on the final is 85.

22. Identify the null and alternative hypotheses for the following: It has been stated that 75 out of every 100 people who go to the movies on Saturday night buy popcorn.

23. The following information was obtained from independent random samples. Assume normally distributed populations with equal variances.

Sample 1 Sample 2

Sample Mean 45 42

Sample Variance 85 90

Sample Size 10 10

The point of estimate for the standard deviation of the difference between the means of the two populations is

a. 3

b. 4.01

c. 9.37

d. 16.09

e. None of the above answers is correct.

24. The critical F value with 6 numerator and 60 denominator degrees of freedom at a = .05 is

a. 3.74

b. 2.25

c. 2.37

d. None of the above answers is correct.

25. The following information was obtained from matched samples.

Individual Method 1 Method 2

1 7 6

2 5 8

3 6 7

4 7 6

5 5 6

The point estimate for the difference between the means of the two populations is

a. -3

b. -0.6

c. 1

d. 6.3

e. None of the above answers is correct.

26. In order to estimate the difference between the average daily sales of two branches of a department store, the following data has been gathered. Assume the two populations are normally distributed and have equal variances.

Downtown Store North Mall Store

Sample size 12 days 14 days

Sample mean $36,000 $32,000

Sample standard deviation $1,200 $1,000

A point estimate for the difference between the two sample means is

a. 2

b. 200

c. 4000

d. 32000

e. None of the above answers is correct.