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6.1 Confidence Intervals for the fd
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Mean (Large Samples)
Mean (Large Samples)
Estimating Population Parameters
Estimating Population Parameters
• Point
Point Estimate is a single value estimate for Estimate is a single value estimate for
the population parameter. Most likely is the sample mean.
• Interval estimate is an interval, or range of value, used to estimate a population parameter
• Level of confidence is the probability that the interval estimate contains the population parameter.
Level of Confidence
Level of Confidence
Level of Confidence
Z‐score
Level of Confidence
Level of Confidence
Level of Confidence
Z‐score
90%
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1.645
Level of Confidence
Level of Confidence
Level of Confidence
Z‐score
90%
1 645
1.645
95%
1.96
Level of Confidence
Level of Confidence
Level of Confidence
Z‐score
90%
1 645
1.645
95%
1.96
99%
2.575
Finding the z‐score
Finding the z
score
Level of confidence is 88%
Level of confidence is 88%
Look on the chart and find .06 Look
on the chart and find .06
and tell the corresponding z‐
score
6%
About 1.555
88%
6%
Finding Point Estimate
Finding Point Estimate
Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. The following represents a random sample of
readability for magazine advertisements. The following represents a random sample of the number of sentences found in 54 advertisements. Find the point estimate of the population mean.
9 20 18 19 9 16 16 9 11 13 22 16 5 18 6 6 5 12
25 17 23 7 10 9 10 10 5 11 18 18 9 9 17 13 11 7
25 17 23 7 10 9 10 10 5 11 18 18 9 9 17 13 11 7 14 6 11 12 11 15 6 12 14 11 4 9 18 12 12 17 11 20
A good point estimate is the sample mean.
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About 12 481
About 12.481
Margin of Error
Margin of Error
Margin of Error, E, is the greatest possible distance between the point estimate and the value of the parameter it is estimating given the level of confidence.
When n is greater than 30, the sample standard deviation, s, can be used in place of sigma.
Finding the Margin of Error
Finding the Margin of Error
Use the data from the first example and a 95% confidence level and a standard deviation of 5 to find the margin of error for the mean number of sentences in all magazine advertisements.
You are 95% confident that the margin of error for the population mean is You
are 95% confident that the margin of error for the population mean is
about 1.333 sentences.
Confidence Interval
Confidence Interval
Construct a Confidence Interval
Construct a Confidence Interval
Construct a 95% confidence interval for the mean number of sentences in all magazine advertisement. advertisement.
From the previous examples With a 95% confidence, you can say that the population mean number of With
a 95% confidence, you can say that the population mean number of
sentences is between 11.148 and 13.814
Guidelines for Constructing a Confidence Interval
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• Find
Find the n, the standard deviation, and the the n the standard deviation and the
sample mean (point of estimate)
• Find the z‐score that corresponds to the given Find the z score that corresponds to the given
level of confidence
• Find the margin of error
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• Find the left and right endpoints of the confidence interval
Constructing a Confidence Interval
Constructing a Confidence Interval
A college admissions director wishes to estimate the mean age of all students currently enrolled In a random sample of 20 students the mean age is found to be
currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age.
Find the z‐score from the chart that corresponds to 90%
Find the margin of error
Find the confidence interval
With 90% confidence, you can say that the mean age of all the
say that the mean age of all the students is between 22.348 and 23.452.
Constructing a Confidence Interval
Constructing a Confidence Interval
Construct a 90% confidence interval of the population mean age for all the college students in the previous example if the sample size is increased to 30 students.
students in the previous example if the sample size is increased to 30 students.
With 90% confidence, you can say that the mean age of all the students is between 22.449 and 23.351
Determine the Minimum Sample Size
Determine the Minimum Sample Size
You want to estimate the mean number of sentences in a magazine advertisement. How many magazine advertisements must be included in the sample if you want to be 95% confident that the sample mean is within one sentence of the population mean. Assume the standard deviation is 5.
So you should include at least 97 magazine advertisements in your sample