From the Generic Syllabus for Chapter 9

The student will be able to:

1. Determine sample size necessary for a specific confidence interval.

2. Estimate the value of a population proportion by the point estimate and confidence interval.

3. Estimate the value of a population mean by the point estimate and confidence interval.

4. Interpret confidence intervals.

5. Use technology to calculate confidence intervals and sample sizes.

Section 9.1 – Estimating a Population Proportion

Objectives

1. Obtain a point estimate for the population proportion

2. Construct and interpret a confidence interval for the population proportion

3. Determine the sample size necessary for estimating the population proportion within a specified margin of error

Objective 1 – Obtain a point estimate for the population proportion

Point Estimate – The value of a statistic that estimates the value of a parameter.

(for example, “I estimate that the proportion of female students enrolled in Statistics is .84”)

The point estimate for a sample proportion is called p-hat and is denoted as .

The population proportion is called rho and is denoted by .

is the point estimate for ρ (x = number of successes, n = # of trials)

Objective 2 - Construct and interpret a confidence interval for the population proportion

Remember from Chapter 7:

is the z score for which the area under the curve to the right of the z score is

Confidence Interval and Level of Confidence

The most common confidence levels are: 90%, 95%, 99%

Note the relationship between confidence level and alpha when filling out the table below.

Margin of Error (E) – A measure of how accurate the point estimate is.

How are confidence intervals created?

To create confidence intervals, take the point estimate from the sample and add and subtract the margin of error.

Point Estimate ± Margin of Error

Note, the Margin of Error of the sample proportion is , given a level of confidence.

The margin of error depends on three factors:

· Level of confidence: As the level of confidence increases, the margin of error also increases.

· Sample size: As the size of the random sample increases, the margin of error decreases.

· Standard deviation of the population: The more spread there is in the population, the wider our interval will be for a given level of confidence.

The value is called the critical value of the distribution. It represents the number of standard deviations the sample statistic can be from the parameter and still result in an interval that includes the parameter.

Fill out the following table

Level of Confidence
(1-)*100% / / Area in each tail, / Critical value,
90%
95%
99%

Example

Determinethat corresponds to a 96% level of confidence.
To calculate a confidence interval by hand

Independence requirement -

Normality requirement -

To calculate a confidence interval with the TI 83/84

Example

A survey found 195 of 250 randomly selected Internet users have high-speed Internet access at home.

a) Calculate a point estimate for the proportion of all internet users who have high-speed internet access at home.

b) Construct a 90% confidence interval for the proportion of all Internet users who have high-speed Internet access at home.

How to Interpret a Confidence Interval

I am __% confident that the true proportion of ______is between __ and __.

A 95% level of confidence does NOT tell us that “there is a 95% probability that the parameter lies between the lower and upper bound”.

A 95% confidence interval means that the method “works” for 95% of all samples. We do not know if our interval is one of the 95% that contain the parameter or the 5% that does not contain the parameter.

Put another way, a 95% level of confidence implies that if 100 different confidence intervals are constructed, each based on a different sample from the same population, then we expect 95 of the intervals to include the parameter and 5 to not include the parameter.

Example

From the internet survey above, (195 of 250 randomly selected Internet users have high-speed Internet access at home), write a sentence to interpret the interval.

Objective 3 - Determine the sample size necessary for estimating the population proportion within a specified margin of error

The following table will save time if you memorize

Confidence Interval Critical Values,
Level of Confidence / Critical Value,
0.90 or 90% / 1.645
0.95 or 95% / 1.96
0.98 or 98% / 2.33
0.99 or 99% / 2.575

Examples

a) An economist wants to know if the proportion of the US population who commute to work by car-pooling is on the rise due to higher gas prices. What sample size should be obtained if the economist wants an estimate within 2 percentage points of the true proportion with 95% confidence?

b) A sociologist wanted to determine the percentage of residents of America that only speak English at home. What size sample should be obtained if she wishes her estimate to be within 3 percentage points with 90% confidence assuming she uses the 2000 estimate obtained from the Census 2000 Supplementary Survey of 82.4%?

Section 9.2 – Estimating a Population Mean

Objectives

1. Obtain a point estimate for the population mean

2. State properties of Student’s t-distribution

3. Determine t-values

4. Construct and interpret a confidence interval for a population mean

Find the sample size needed to estimate the population mean within a given margin or error

Objective 1 – Obtain a point estimate for the population mean

A point estimate is the value of a statistic that estimates the value of a parameter.

The sample mean is a point estimate of the population mean μ.

Example

Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent the weights (in grams) of 17 randomly selected pennies minted after 1982.

2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.45 2.49 2.47 2.45 2.46 2.45 2.46 2.47 2.44 2.45

Treat the data as a simple random sample. Estimate the population mean weight of pennies minted after 1982.

Objective 2 – State the properties of the Student’s t-distribution

Use the t-distribution when σ is unknown and EITHER n > 30 or the population is bell-shaped.

Objective 4 – Construct and interpret a confidence interval for a population mean

As with the population proportion, the confidence interval will be

Point estimate margin of error

Example

Construct a 99% confidence interval about the population mean weight (in grams) of pennies minted after 1982. Assume μ = 0.02 grams.

2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.45 2.49 2.47 2.45 2.46 2.45 2.46 2.47 2.44 2.45

Write a sentence to interpret the interval.

Example

In a study of the length of time that it takes students to earn a bachelor’s degree, 80 students are randomly selected and they are found to have a mean of 4.8 years and a standard deviation of 2.2 years. Construct a 95% confidence interval estimate of the mean amount of time it takes all students to earn a bachelor’s degree. Write a sentence to interpret the interval.

Objective 5 - Find the sample size needed to estimate the population mean within a given margin or error

Back to the pennies. How large a sample would be required to estimate the mean weight of a penny manufactured after 1982 within 0.005 grams with 99% confidence? Assume s = 0.02.