Question

 

Question 1

What is the point estimate for a population mean? Provide examples.

Question 2

What are the confidence intervals for a population mean? Provide examples.

Question 3

A confidence interval for the population mean when the population follows the normal distribution and the population standard deviation is known is computed by? Provide examples.

Answer 1

Question 1

What is the point estimate for a population mean?

Ans.: The point estimate for a population mean () is the sample mean (x-bar).

Provide examples.

Ans.:

Suppose we want to find out the average weight of all students in 10th Grade in a particular school. We randomly select ten students and weigh them. (randomly as in, by using some method of random sampling) The mean weight of the sample of students is found to be 55 Kg, so that number is our point estimate(of the population mean weight, in this case).

Question 2

What are the confidence intervals for a population mean?

Ans.: We compute the sample mean in order to estimate the population mean. However some error is associated with this estimate since the true population mean may be different (larger or smaller) than the sample mean. Instead of a point estimate, we may identify a range of possible values such that, ? (population mean) will not be lower than the lowest value in this range and not higher than the highest value. Such a range is called a confidence interval.

Now we may compute, say, a 95% Confidence Interval for the population mean. This would mean that if repeated samples were taken and the 95% confidence interval was computed for each sample, 95% of the intervals would contain the population mean, i.e., a 95%confidence interval has a 0.95 probability of containing the population mean.

Provide examples.

Ans.:

Suppose we want to estimate, with 95% confidence, the mean (average) height of all students in 10th Grade in a particular school. We take a random sample of 10 students and determine that the average height is 150 cm and the sample standard deviation is 5 cm. So the Confidence Interval would be: (150 +- 2.89) i.e. the lower end of the interval is 150 - 2.89 = 147.11 cm and the upper end is 150 + 2.89 = 152.89 cm. Can be written as (147.11, 152.89)

Question 3

A confidence interval for the population mean when the population follows the normal distribution and the population standard deviation is known is computed by?

Ans.: If the population follows the normal distribution and the population standard deviation is known, to construct a confidence interval for the population mean, we need the point estimate of the population mean i.e.

The confidence interval estimate will have the form:

Calculating the Confidence Interval

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:

• Calculate the sample mean from the sample data.
• Find the z-score that corresponds to the confidence level.
• Calculate the error bound.
• Construct the confidence interval.
• Interpret.

Provide examples.

Ans.:

To construct a 95% confidence interval for the population mean, when the population follows the normal distribution and the population standard deviation is known:

Suppose we want to estimate, with 95% confidence, the mean (average) income of all people living in a particular region. We take a random sample of 30 people and determine that the average income is 30000 and the sample standard deviation is 10000.

Because we want a 95% confidence interval, we determine the t*-value as follows:

The t*-value comes from a t-distribution with 30 – 1 = 29 degrees of freedom. This t*-value is found by looking at the t-table. We find t* = 2.045. This is the t*-value for a 95% confidence interval for the mean with a sample size of 30.

We multiply 2.045 times 10000 divided by the square root of 30. The margin of error is, therefore,

+- 3733.64

Therefore the 95% confidence interval for the mean income of all people living in a particular region is

(30000 +- 3733.64) or (26266.36, 33733.64)