Confidence Intervals for a Mean

when you have a “large” sample…

The situation

• Want to estimate the actual population mean .

• But can only get , the sample mean.

• Find a range of values, L < < U, that we can be really confident contains .

• This range of values is called a “confidence interval.”

Confidence Intervals for Proportions in Newspapers

• 18% of women, aged 18-24, think they are overweight.

• The “margin of error” is 5%.

• The “confidence interval” is 18% ± 5%.

• We can be really confident that between 13% and 23% of women, aged 18-24, think they are overweight.

General Form of most Confidence Intervals

• Sample estimate ± margin of error

• Lower limit L = estimate - margin of error

• Upper limit U = estimate + margin of error

• Then, we’re confident that the population value is somewhere between L and U.

Example

• Let X = number of high school friends Stat 250 students keep in touch with.

• True population mean = 5 friends.

• True population standard deviation = 5 friends.

• Take a random sample of 36 Stat 250 students. Calculate .

Sampling Distribution of

Sample means

5 5 + 2(0.83) 6.7

5 - 2(0.83)3.3

0.95

0.83365

nσ

66.1)83.0(2)n

σ(2

What does the sampling distribution tell us?

• 95% of the sample means will fall within 2 standard errors, or within 1.66 friends, of the true population mean = 5.

• Or, 95% of the time, the true population mean = 5 will fall within 2 standard errors, or within 1.66 friends, of the sample mean.

Sampling Distribution of

Sample means

+ 2(/n) - 2(/n)

0.95

)n

σ(2

What does the sampling distribution tell us?

• 95% of the sample means will fall within 2 standard errors of the population mean

• Or, 95% of the time, the true population mean will fall within 2 standard errors of the sample mean.

• Use this last statement to create a formula.

95% Confidence Interval for

n

σ2X

Formula in notation:

Formula in English:

Sample mean ± (2 × standard error of the mean)

95% Confidence Interval for

ns2XFormula in notation:

Formula in English:

Sample mean ± (2 × estimated standard error)

1. Formula OK as long as sample size is large (n 30)

2. Margin of error = 2 × standard error of the mean

3. 95% is called the “confidence level”

Example

• A random sample of 32 students reported combing their hair an average of 1.6 times a day with a standard deviation of 1.3 times a day.

• In what range of values can we be 95% confident that , the actual mean, falls?

What does 95% confident mean?

Sample means

+ 2(/n) - 2(/n)

0.95

)n

σ(2

95% of all such confidence intervals will contain the true mean

What if you want to be more (or less) confident?

Sample means

+ Z(/n) - Z(/n)

0.98

1. Put confidence level in middle.

2. Subtract from 1.3. Divide by 2 and put

in tails.4. Look up Z value.

0.010.01

Z0.99 = 2.33

Any % Confidence Interval for

nsX ZFormula in notation:

Formula in English:

Sample mean ± (Z × estimated standard error)

Example

• A random sample of 64 students reported having an average of 2.4 roommates with a standard deviation of 4 roommates.

• In what range of values can we be 96% confident that , the actual mean, falls?

Length of Confidence Interval

• Want confidence interval to be as narrow as possible.

• Length = Upper Limit - Lower Limit

How length of CI is affected?

• As sample mean increases…

• As the standard deviation decreases…

• As we decrease the confidence level…

• As we increase sample size …

nsX Z

Warning #1

• Confidence intervals are only appropriate for random, representative samples.

• Problematic samples: – magazine surveys– dial-in surveys (1-900-vote-yes)– internet surveys (CNN QuickVote)

Warning #2

• The confidence interval formula we learned today is only appropriate for large samples (n 30).

• If you use today’s formula on a small sample, you’ll get a narrower interval than you should.

• Will learn correct formula for small samples.

Warning #3

• The confidence interval for the mean is a range of possible values for the population average.

• It says nothing about the range of individual measurements. The empirical rule tells us this.