Chapter 8Confidence 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 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.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:

EXAMPLE Using Simulation to Demonstrate the Idea of a Confidence Interval

We will use Minitab to simulate obtaining 60 simple random samples of size n = 8 from a population that is normally distributed with = 50 and = 10. Construct a 95% confidence interval for each sample. How many of the samples result in intervals that contain = 50.

Sample Mean 95.0% CI C1 47.07 ( 40.14, 54.00)C2 49.33 ( 42.40, 56.26)C3 50.62 ( 43.69, 57.54)C4 47.91 ( 40.98, 54.84)C5 44.31 ( 37.38, 51.24)C6 51.50 ( 44.57, 58.43)C7 52.47 ( 45.54, 59.40)C8 59.62 ( 52.69, 66.54)C9 43.49 ( 36.56, 50.42)C10 55.45 ( 48.52, 62.38)C11 50.08 ( 43.15, 57.01)C12 56.37 ( 49.44, 63.30)

SAMPLE MEAN 95% CIC13 49.05 ( 42.12, 55.98)C14 47.34 ( 40.41, 54.27)C15 50.33 ( 43.40, 57.25)C16 44.81 ( 37.88, 51.74)C17 51.05 ( 44.12, 57.98)C18 43.91 ( 36.98, 50.84)C19 46.50 ( 39.57, 53.43)C20 49.79 ( 42.86, 56.72)C21 48.75 ( 41.82, 55.68)C22 51.27 ( 44.34, 58.20)C23 47.80 ( 40.87, 54.73)C24 56.60 ( 49.67, 63.52)C25 47.70 ( 40.77, 54.63)C26 51.58 ( 44.65, 58.51)C27 47.37 ( 40.44, 54.30)C28 61.42 ( 54.49, 68.35)

SAMPLE MEAN 95% CIC29 46.89 ( 39.96, 53.82)C30 51.92 ( 44.99, 58.85)C31 48.57 ( 41.64, 55.49)C32 51.49 ( 44.57, 58.42)C33 54.89 ( 47.96, 61.82)C34 45.11 ( 38.18, 52.04)C35 47.90 ( 40.97, 54.83)C36 50.03 ( 43.10, 56.96)C37 48.16 ( 41.23, 55.09)C38 54.06 ( 47.13, 60.99)C39 53.55 ( 46.62, 60.48)C40 50.39 ( 43.46, 57.32)C41 50.78 ( 43.85, 57.71)C42 53.44 ( 46.51, 60.37)C43 48.47 ( 41.54, 55.40)C44 46.11 ( 39.18, 53.04)

SAMPLE MEAN 95% CIC45 52.61 ( 45.68, 59.54)C46 46.74 ( 39.81, 53.67)C47 48.13 ( 41.20, 55.06)C48 44.47 ( 37.54, 51.40)C49 48.96 ( 42.03, 55.89)C50 53.43 ( 46.50, 60.36)C51 50.94 ( 44.01, 57.87)C52 53.42 ( 46.49, 60.35)C53 49.87 ( 42.94, 56.80)C54 51.31 ( 44.38, 58.24)C55 51.71 ( 44.78, 58.64)C56 53.80 ( 46.87, 60.73)C57 45.70 ( 38.77, 52.63)C58 49.80 ( 42.87, 56.73)C59 49.81 ( 42.88, 56.73)C60 52.87 ( 45.94, 59.80)

EXAMPLE Constructing a Confidence Interval

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

Weight (in grams) of Pennies

EXAMPLE Role of Margin of Error

Construct a 90% confidence interval for the mean weight of pennies minted after 1982. 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