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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.