Chapter 8 Confidence Intervals 8.1 Confidence Intervals about a Population Mean,  Known

Appropriately obtaining individuals to participate in a survey or appropriate design of an experiment is vital for the statistical process to be valid. In other words, if data is carelessly or inappropriately collected, any statistical inference performed on the data is subject to scrutiny. For example, the results of Internet surveys should be looked upon with extreme skepticism.

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

EXAMPLE Computing a Point Estimate for the Population Mean 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 Treat the data as a simple random sample. Estimate the population mean weight of pennies minted after Specifications call for the penny to weigh 2.50 grams. Is there reason to believe the pennies are minted to specification?

A confidence interval estimate of a parameter consists of an interval of numbers along with a probability that the interval contains the unknown parameter.

The level of confidence in a confidence interval is a probability that represents the percentage of intervals that will contain if a large number of repeated samples are obtained. The level of confidence is denoted

For example, a 95% level of confidence would mean that if 100 confidence intervals were constructed, each based on a different sample from the same population, we would expect 95 of the intervals to contain the population mean.

The construction of a confidence interval for the population mean depends upon three factors  The point estimate of the population  The level of confidence  The standard deviation of the sample mean

Suppose we obtain a simple random sample from a population. Provided that the population is normally distributed or the sample size is large, the distribution of the sample mean will be normal with

95% of all sample means are in the interval With a little algebraic manipulation, we can rewrite this inequality and obtain:

Sample Mean 95.0% CI C ( 40.14, 54.00) C ( 42.40, 56.26) C ( 43.69, 57.54) C ( 40.98, 54.84) C ( 37.38, 51.24) C ( 44.57, 58.43) C ( 45.54, 59.40) C ( 52.69, 66.54) C ( 36.56, 50.42) C ( 48.52, 62.38) C ( 43.15, 57.01) C ( 49.44, 63.30)

SAMPLEMEAN95% CI C ( 42.12, 55.98) C ( 40.41, 54.27) C ( 43.40, 57.25) C ( 37.88, 51.74) C ( 44.12, 57.98) C ( 36.98, 50.84) C ( 39.57, 53.43) C ( 42.86, 56.72) C ( 41.82, 55.68) C ( 44.34, 58.20) C ( 40.87, 54.73) C ( 49.67, 63.52) C ( 40.77, 54.63) C ( 44.65, 58.51) C ( 40.44, 54.30) C ( 54.49, 68.35)

SAMPLE MEAN95% CI C ( 39.96, 53.82) C ( 44.99, 58.85) C ( 41.64, 55.49) C ( 44.57, 58.42) C ( 47.96, 61.82) C ( 38.18, 52.04) C ( 40.97, 54.83) C ( 43.10, 56.96) C ( 41.23, 55.09) C ( 47.13, 60.99) C ( 46.62, 60.48) C ( 43.46, 57.32) C ( 43.85, 57.71) C ( 46.51, 60.37) C ( 41.54, 55.40) C ( 39.18, 53.04)

SAMPLEMEAN95% CI C ( 45.68, 59.54) C ( 39.81, 53.67) C ( 41.20, 55.06) C ( 37.54, 51.40) C ( 42.03, 55.89) C ( 46.50, 60.36) C ( 44.01, 57.87) C ( 46.49, 60.35) C ( 42.94, 56.80) C ( 44.38, 58.24) C ( 44.78, 58.64) C ( 46.87, 60.73) C ( 38.77, 52.63) C ( 42.87, 56.73) C ( 42.88, 56.73) C ( 45.94, 59.80)

EXAMPLEConstructing a Confidence Interval

Weight (in grams) of Pennies

EXAMPLERole of Margin of Error Construct a 90% confidence interval for the mean weight of pennies minted after Comment on the effect decreasing the level of confidence has an the margin of error.

EXAMPLE The Role of Sample Size on the Margin of Error

Determining the Sample Size n

EXAMPLE Determining Sample Size