Chapter 9.1 Estimating a Population Proportion Objective A : Point Estimate A point estimate is the value of a statistic that estimates the value of a parameter.

x The best point estimate of the population proportion is a sample proportion ( p = ). n ∑ x ). The best point estimate of the population mean is a sample mean ( x = n Since p varies from sample to sample, we use an interval based on p to capture the unknown population proportion with a level of confidence.

Objective B : Confidence Interval A confidence interval for an unknown parameter consists of an interval of numbers based on a point estimate. The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained.

The level of confidence is denoted as (1 − α ) ⋅100% . The level of confidence controls the width of the interval. Confidence interval estimates for a parameter are of the form: Point estimate ± margin of error. Confidence interval for p :

p (1 − p ) provided that n p (1 − p ) ≥ 10 n The value of Zα /2 is called the critical value of the distribution. p ± Z ⋅ σ p α /2

where σ p =

Example 1: Determine the critical value Zα /2 that corresponds to the given level of confidence. (a) 90%

(b) 95%

(c) 98%

(d) 92%

The margin of error, E , in a (1 − α ) ⋅100% confidence interval for a population proportion is given by E = Zα /2

p (1 − p ) . The width of the interval is determined by the margin of error. n

Example 2: Determine the margin of error for p with x = 540 and n = 900 at a 99% level of confidence.

Example 3: Construct a confidence interval of the population proportion at the given level of confidence. = x 80, = n 200, 96% confidence

Objective C : Sample Size Needed for Estimating the Population Proportion p The sample size required to obtain a (1 − α ) ⋅100% confidence interval for p with a margin of error E is given by Z  = n p (1 − p )  α /2   E 

2

Round up to the next integer

p is a prior estimate of p If a prior estimate of p is unavailable, the sample size required is Z  n = 0.25  α /2   E 

2

Round up to the next integer

Example 1 : An urban economist wishes to estimate the proportion of Americans who own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with 90% confidence if (a) he uses a 2010 estimate of 0.669 obtained from the U.S Census Bureau?

(b) he does not use any prior estimates?

Example 2: In a Gallup poll conducted in October 2010, 64% of the people polled answered "more strict" to the following question: "Do you feel that the laws covering the sale of firearms should be made more strict as they are now?" Suppose the margin of error in the poll was 3.5% and the estimate was made with 95% confidence. At least how many people were surveyed?

Example 3: A Gallup poll conducted in November 2010 found that 493 of 1050 adult Americans believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. (a) Obtain a point estimate for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage.

(b) Verify the requirements for constructing a confidence interval for p are satisfied.

(c) Construct a 95% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans healthcare coverage.

(d) Construct a 99% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage.

(e) What is the effect of increasing the level of confidence on the width of the interval?

Chapter 9.2 Estimating a Population Mean Objective A : Point Estimate The best point estimate of the population mean, µ , is the sample mean, x . Objective B : Student's t - distribution Properties of the t - distribution 1. The t - distribution is different for different degrees of freedom ( df = n − 1 ). 2. The t - distribution has the same general symmetric bell shape as the standard normal distribution but its area in the tails is a little greater than the area in the tails of the standard normal distribution due to the greater variability that is expected with small samples. 3. The t - distribution has a mean of t = 0 at the center of the distribution.

4. As the sample size n gets larger, the t - distribution gets closer to the standard normal distribution.

Example 1: Determine the t -value. (a) Find the t -value such that the area in the right tail is 0.05 with 19 degrees of freedom.

(b) Find the t -value such that the area left of the t -value is 0.02 with 6 degrees of freedom. [Hint: Use symmetry.]

(c) Find the critical t -value that corresponds to 90% confidence. Assume 12 degrees of freedom.

In general, the population standard deviation is unknown for estimating a population mean based on a sample mean. The t -distribution is used to off-set the additional variability introduced by using s in place of σ .

Objective C : Confidence Interval for a Population Mean

Constructing a (1 − α ) ⋅100% Confidence Interval for µ Point estimate ± margin of error

s n provided the data come from a population that is normally distributed, or the sample size is large. x ± tα /2 ⋅

Example 1: A simple random sample of size n < 30 has been obtained. From the normal probability plot and boxplot, judge whether a t -interval should be constructed. (a)

(b)

Example 2 : A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x , is found to be 50, and the sample standard deviation, s , is found to be 8. (a) Construct a 98% confidence interval for µ if the sample size, n , is 20.

(b) Construct a 98% confidence interval for µ if the sample size, n , is 15. How does decreasing the sample size affect the margin of error, E ?

(c) Construct a 95% confidence interval for µ if the sample size, n , is 20. Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, E ?

(d) Could we have computed the confidence intervals in parts (a) to (c) if the population had not been normally distributed? Why?

Example 3 : How much time do Americans spend eating or drinking? Suppose for a random sample of 1001 Americans age 15 or older, the mean amount of time spent eating or drinking per day is 1.22 hours with a standard deviation of 0.65 hour. (a) A histogram of time spent eating and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day.

(b) Determine and interpret a 95% confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day.

(c) Could the interval be used to estimate the mean amount of time a 9-year-old American spends eating and drinking each day? Explain.

Objective D : Determining the Sample Size n The sample size required to estimate the population mean, µ , with a level of confidence

(1 − α ) ⋅100%

within a specified margin of error, E , is given by

 Z ⋅s  n =  α /2   E  where n is rounded up to the nearest whole number. 2

Note: The t -distribution approaches the standard normal z - distribution as the sample size increases.

Example 1: A researcher wanted to determine the mean number of hours per week (Sunday through Saturday) the typical person watches television. Results from the Sullivan Statistics Survey indicate that s = 7.5 hours. (a) How many people are needed to estimate the number of hours people watch television per week within 2 hours with 95% confidence?

(b) How many people are needed to estimate the number of hours people watch television per week within 1 hour with 95% confidence?

(c) What effect does doubling the required accuracy have on the sample size?