1. The estimator Y is a random variable that varies with different random sample
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Question
1. The estimator Y is a random variable that varies with different random samples; it has a probability distribution function that represents its sampling distribution, and mean and variance. Using the properties on expected values and variances of linear functions of random variables and sum operators
The College Board reported the following mean scores for the three parts of the Scholastic Aptitude Test (SAT)
Critical Reading 502 Mathematics 515 Writing 494
3. Assume the population standard deviation on each part of the test is 10
a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test?
c. What is the probability that a random sample of 100 test takers will provide a sample mean score within 10 of the population mean of 494 on the writing part of the test? Comment on the differences between this probability and the values computed in part (a) and (b).
4. Trading volume on the New York Stock exchange is heaviest during the first half hour and last half hour of the trading day. The early morning trading volume (millions of shares) for 13 days in January and February are shown in the table below.
214 163 265 194 180 202 198 212 201 174 171 211 211 The computed sample's mean 200 x , and the sample standard deviation is 26.04
. The probability distribution of trading volume is approximately normal.
a. What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares?
b. What is the probability that, on a randomly selected day, the early trading volume will exceed 230 shares?
c. How many shares would have to be traded for the early morning trading volume on a particular day to be among the busiest 5% of days?
5. The mean number of hours of flying time for pilots at Continental Airlines is 49 hours per month. Assume that this mean was based on actual flying times for a sample of 100 Continental Pilots and that the sample standard deviation was 8.5 hours.
a. At 95% confidence, what is the margin of error?
b. What is the 95% confidence interval estimate of the population mean flying time for the pilots?
6. For the United States, the mean monthly Internet bill is $32.79 per household. A sample of 50 households in a southern state showed a sample mean of $30.63. Use a population standard deviation of 60 .5
a. Formulate hypotheses for a test to determine whether the sample data support the conclusions that the mean monthly Internet bill in the southern state is less than the national mean of $32.79.
b. What is the value of the test statistic?
c. What is the p-value?
8. The University rule of thumb is that students spend 2 hours outside of class per week for every unit of course credit. For a 4 unit class, this means that students should spend 8 hours a week studying for the course. A random sample of 8 students has been selected and asked how many hours per week they spend on the 4 unit course. The sample values are:{ 11, 2, 2, 8, 4, 6, 4, 3. }. The sample mean of 5 and the sample standard deviation sqrt(10)
a. Construct a 95% confidence interval for the population mean number of hours studied per week in this course (assuming the population is normally distributed).
b. State and test a null hypothesis that mean study study-time is 8 hours per week. Use a 5% significance level and perform a two-sided test. Explain why you reject or do not reject your null hypothesis.
c. Test the same hypothesis as in part b using a one-sided test (again with significance level 5 percent).
Explanation / Answer
Answer:
1. The estimator Y is a random variable that varies with different random samples; it has a probability distribution function that represents its sampling distribution, and mean and variance. Using the properties on expected values and variances of linear functions of random variables and sum operators
The College Board reported the following mean scores for the three parts of the Scholastic Aptitude Test (SAT)
Critical Reading 502 Mathematics 515 Writing 494
Assume the population standard deviation on each part of the test is 10
a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
Standard error = sd/sqrt(n) =10/sqrt(90) = 1.0541
Z value for difference 10 from mean = 10/1.0541 =9.49
P(within 10 points of the population mean of 502) = P( -9.49<z<9.49) =1.000
b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test?
Standard error = sd/sqrt(n) =10/sqrt(90) = 1.0541
Z value for difference 10 from mean = 10/1.0541 =9.49
P(within 10 points of the population mean of 515) = P( -9.49<z<9.49) =1.000
c. What is the probability that a random sample of 100 test takers will provide a sample mean score within 10 of the population mean of 494 on the writing part of the test?
Standard error = sd/sqrt(n) =10/sqrt(100) = 1
Z value for difference 10 from mean = 10/1 =10
P(within 10 points of the population mean of 502) = P( -10<z<10) =1.000
Comment on the differences between this probability and the values computed in part (a) and (b).
Since z values are large , both probabilities are same.
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