Confidence Interval Estimation for a Population Mean Lecture 33 Section 10.3 Tue, Nov 14, 2006

Confidence Interval Confidence Interval Estimation for a Estimation for a Population MeanPopulation Mean

Lecture 33Lecture 33

Section 10.3Section 10.3

Tue, Nov 14, 2006Tue, Nov 14, 2006

Confidence IntervalsConfidence Intervals

To estimate To estimate , we will use , we will use confidence intervals, as we did when confidence intervals, as we did when estimating estimating pp..

The basic form, as well as the The basic form, as well as the theory, is the same:theory, is the same:

(pt. est.) (pt. est.) (approp. no. of st. devs.) (approp. no. of st. devs.)


What is the point estimate for What is the point estimate for ?? What is the standard deviation for What is the standard deviation for

this estimator?this estimator? How do we determine the How do we determine the

appropriate number of standard appropriate number of standard deviations?deviations?


IfIfxx has a normal distribution, then has a normal distribution, then the confidence interval isthe confidence interval is

oror

If (If (xx – – )/()/(ss//nn) has a ) has a tt distribution, distribution, then the confidence interval isthen the confidence interval is

nzx /

nszx /

nstx /

When to Use When to Use ZZ

IfIf The population is normal (or nearly The population is normal (or nearly

normal) and normal) and is known, is known, oror The population is not normal, but the The population is not normal, but the

sample size is at least 30,sample size is at least 30, Then use Then use ZZ..

When to Use When to Use tt

IfIf The population is normal (or nearly The population is normal (or nearly

normal), normal), andand is not known, is not known, andand The sample size is less than 30,The sample size is less than 30,

Then use Then use tt..

ExampleExample

Example 10.4, p. 641: The Kellogg Example 10.4, p. 641: The Kellogg Corporation controls approximately a 43% Corporation controls approximately a 43% share of the ready-to-eat cereal market share of the ready-to-eat cereal market worldwide. A popular cereal is Corn Flakes. worldwide. A popular cereal is Corn Flakes. Suppose the weights of full boxes of a certain Suppose the weights of full boxes of a certain kind of cereal are normally distributed with a kind of cereal are normally distributed with a population standard deviation of 0.29 ounces. population standard deviation of 0.29 ounces. A random sample of 25 boxes produced a A random sample of 25 boxes produced a mean weight of 9.82 ounces. mean weight of 9.82 ounces.

Construct a 95% confidence interval for the Construct a 95% confidence interval for the true mean weight of such boxes.true mean weight of such boxes.

ExampleExample

Use Use ZZ. Why?. Why? nn = 25. = 25. xx = 9.82. = 9.82. Assume that Assume that = 0.29. = 0.29. Level of confidence = 95%, so Level of confidence = 95%, so zz = =

1.96.1.96.

ExampleExample

The confidence interval isThe confidence interval is

9.82 9.82 (1.96)(0.29/ (1.96)(0.29/25) 25)

= 9.82 = 9.82 0.114 0.114

= (9.706, 9.934).= (9.706, 9.934).

TI-83 – Confidence TI-83 – Confidence IntervalsIntervals

When the standard normal When the standard normal distribution applies, do the distribution applies, do the following.following.

Press STAT.Press STAT. Select TESTS.Select TESTS. Select ZInterval.Select ZInterval. A window appears requesting A window appears requesting

information.information.


Select Data or Stats.Select Data or Stats. Assume we selected Stats.Assume we selected Stats. Enter Enter .. EnterEnterxx.. Enter Enter nn.. Enter the level of confidence.Enter the level of confidence. Select Calculate and press ENTER.Select Calculate and press ENTER.


A window appears containingA window appears containing The title “ZInterval”.The title “ZInterval”. The confidence interval in interval The confidence interval in interval

notation.notation. The sample mean.The sample mean. The sample size.The sample size.

ExampleExample Example 10.5, p. 643: Unoccupied seats on Example 10.5, p. 643: Unoccupied seats on

flights cause airlines to lose revenue. Suppose flights cause airlines to lose revenue. Suppose that a large airline wants to estimate its that a large airline wants to estimate its average number of unoccupied seats per flight average number of unoccupied seats per flight from Detroit to Minneapolis over the past from Detroit to Minneapolis over the past month. To accomplish this, the records of 61 month. To accomplish this, the records of 61 such flights were randomly selected, and the such flights were randomly selected, and the number of unoccupied seats was recorded for number of unoccupied seats was recorded for each of the sampled flights. The sample mean each of the sampled flights. The sample mean is 12.6 and sample standard deviation is 4.4 is 12.6 and sample standard deviation is 4.4 seats.seats.

Construct a 99% confidence interval for the Construct a 99% confidence interval for the mean number of unoccupied seats.mean number of unoccupied seats.

ExampleExample

Should we use Should we use ZZ or or tt? Why?? Why? nn = 61. = 61. xx = 12.6. = 12.6. ss = 4.4. = 4.4. Level of confidence = 99%. Find Level of confidence = 99%. Find tt..

ExampleExample

Consider again the Consider again the tt table (Table IV). table (Table IV). The degrees of freedom include every The degrees of freedom include every

value up to 30, then jump to 40, 60, value up to 30, then jump to 40, 60, 120.120.

If the actual degrees of freedom are If the actual degrees of freedom are Between 30 and 40, use 30.Between 30 and 40, use 30. Between 40 and 60, use 40.Between 40 and 60, use 40. Between 60 and 120, use 60.Between 60 and 120, use 60.

If they are beyond 120, use If they are beyond 120, use zz..

ExampleExample

The confidence interval isThe confidence interval is

12.6 12.6 (2.660)(4.4/ (2.660)(4.4/61) 61)

= 12.6 = 12.6 1.499 1.499

= (11.101, 14.099).= (11.101, 14.099).


To use To use tt, do the following., do the following. Press STAT.Press STAT. Select TESTS.Select TESTS. Select TInterval.Select TInterval. A window appears requesting A window appears requesting

information.information.


Select Data or Stats.Select Data or Stats. Assume we selected Stats.Assume we selected Stats. EnterEnterxx.. Enter Enter ss.. Enter Enter nn.. Enter the level of confidence.Enter the level of confidence. Select Calculate and press ENTER.Select Calculate and press ENTER.


A window appears containingA window appears containing The title “TInterval”.The title “TInterval”. The confidence interval in interval The confidence interval in interval

notation.notation. The sample mean.The sample mean. The sample standard deviation.The sample standard deviation. The sample size.The sample size.

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Confidence Interval Estimation for a Population Mean Lecture 33 Section 10.3 Tue, Nov 14, 2006