MATH 2190-The Central Limit Theorem and Z-procedures 1 Sampling Distributions

MATH 2190-The Central Limit Theorem and Z-procedures 1 Sampling Distributions

Subject: Mathematics    / Statistics
Question
Homework 4: The Central Limit Theorem and Z-procedures 1 Sampling Distributions 1.1 Apply the Central Limit Theorem According to the 1993 World Factbook, the 1993 total fertility rate (mean
number of children born per woman) for Madagascar is 6.75. Suppose the
standard deviation of the total fertility rate is 2.5. The mean number of
children for a sample of 200 randomly selected women is one value of many
that form the sampling distribution of sample means.
1. What is the shape of this sampling distribution?
2. What is the mean value for this sampling distribution?
3. What is the standard deviation of this sampling distribution? 1.2 Application of the Central Limit Theorem with
Probabilities The waiting time at a certain bank is approximately normally distributed
with a mean of 3.7 min and a standard deviation of 1.4 min.
1. Find the probability that a randomly selected customer has to wait less
than 2.0 min.
2. Find the probability that a randomly selected customer has to wait
more than 6.0 min.
3. If a random sample of 2 customers is selected, find the probability that
the average waiting time is more than 6.0 min.
4. If a random sample of 4 customers is selected, find the probability that
the average waiting time is more than 6.0 min.
5. If a random sample of 10 customers is selected, find the probability
that the average waiting time is more than 6.0 min. 1 2 Theory
1. State The Central Limit Theorem.
2. Define p-value.
3. Define Statistical Significance. In order to perform these type of procedures, we must make certain that
certain conditions have been met. Please list these three conditions:
•
•
• 3 Test of Hypothesis It is known that the mean income for all assembly-line workers with less than
5 years of experience in a large compay is $300 per week. A representative of
a women’s group believes that the female employees are being underpaid. A
random sample of sixteen female employees yields an average weekly income
of $270. Conduct a statistical test that will give you evidence about whether
the female employees have a mean income of less than $300 per week. It
is assumed that the weekly incomes for all the female employees follows a
Normal Distribution with standard deviation ? = $36.
1. What is the population of interest?
2. What is the sample?
3. What is the sample design?
4. Conduct the test:
• State the Null and Alternative Hypothesis:
H0 :
Ha :
• Find these values from the story:
x¯ =
n=
?= 2 • What does the Central Limit Theorem say under the assumption
that the Null is true?
Draw the Distribution and mark off your sample mean.
• Compute the Test Statistic:
z = x¯??µ
?
n • Calculate the p-value
• Do you Reject the H0 or Fail to Reject the Ho ?
• Write out a statement describing your conclustion. 4 Confidence Interval High school students who take the SAT mathematics exam a second time
generally score higher than on their first attempt. The change in score has a
Normal distribution with a standard deviation ? = 50. A random sample of
1000 students gain an average of x¯ = 22 points on their second try.
1. Compute a 95% confidence interval for the mean score gain µ in the
population of all students who take the test a second time.
2. Will the confidence interval become larger or smaller if we descrease
the level of confidence to say 90%?
3. Will the confidence interval become larger or smaller if we increase our
sample size. Why?
4. How large of a sample of high school students in the above example
would be needed to estimate the mean change µ in SAT scores to be
within ±1.5 points with 95%. 3