Confidence Interval for a Mean

when you have a “small” sample...

As long as you have a “large” sample….

A confidence interval for a population mean is:

n

sZx

where the average, standard deviation, and n depend on the sample, and Z depends on the confidence level.

Example

Random sample of 59 students spent an average of $273.20 on Spring 1998 textbooks. Sample standard deviation was $94.40.

09.2420.27359

4.9496.120.273

We can be 95% confident that the average amount spent by all students was between $249.11 and $297.29.

What happens if you can only take a “small” sample?

• Random sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.

• What is the average amount all students slept last night?

If you have a “small” sample...

Replace the Z value with a t value to get:

n

stx

where “t” comes from Student’s t distribution, and depends on the sample size through the degrees of freedom “n-1”.

Student’s t distribution versus Normal Z distribution

-5 0 5

0.0

0.1

0.2

0.3

0.4

Value

dens

ity

T-distribution and Standard Normal Z distribution

T with 5 d.f.

Z distribution

T distribution

• Shaped like standard normal distribution (symmetric around 0, bell-shaped).

• But, t depends on the degrees of freedom “n-1”.

• And, more likely to get extreme t values than extreme Z values.

Graphical Comparison of T and Z Multipliers

0.90 0.92 0.94 0.96 0.98 1.00

0

1

2

3

4

5

Cumulative Probability

Z o

r T

Mul

tiplie

r T with 5 df

Z distribution

Tabular Comparison of T and Z Multipliers

Confidencelevel

t value with5 d.f

Z value

90% 2.015 1.65

95% 2.571 1.96

99% 4.032 2.58

For small samples, T value is larger than Z value.

So,T interval is made to be longer than Z interval.

Back to our example!

Sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.

55.04.615

1145.24.6

n

stx

Need t with n-1 = 15-1 = 14 d.f. For 95% confidence, t14 = 2.145

That is...

We can be 95% confident that average amount slept last night by all students is between 5.85 and 6.95 hours.

Hmmm! Adults need 8 hours of sleep each night.

Logical conclusion:On average, students need more sleep.

(Just don’t get it in this class!)

T-Interval for Mean in Minitab

T Confidence Intervals

Variable N Mean StDev SE Mean 95.0 % CIComb 89 2.011 1.563 0.166 (1.682, 2.340)

We can be 95% confident that the average number of times a “Stat-250-like” student combs his/her is between 1.7 and 2.3 times a day.

T- interval in Minitab

• Select Stat.

• Select Basic Statistics.

• Select 1-Sample t…

• Select desired variable.

• Specify desired confidence level.

• Say OK.

What happens as sample gets larger?

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0.0

0.1

0.2

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Value

dens

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T-distribution and Standard Normal Z distribution

Z distribution

T with 60 d.f.

What happens to CI as sample gets larger?

n

sZx

n

stx

For large samples:

Z and t values become almost identical, so CIs will be almost identical.

Example

Random sample of 64 students spent an average of 3.8 hours on homework last night with a sample standard deviation of 3.1 hours.

Z Confidence Intervals The assumed sigma = 3.10

Variable N Mean StDev 95.0 % CIHomework 64 3.797 3.100 (3.037, 4.556)

T Confidence IntervalsVariable N Mean StDev 95.0 % CIHomework 64 3.797 3.100 (3.022, 4.571)

One not-so-small problem!

• It is only OK to use the t interval for small samples if your original measurements are normally distributed.

• We’ll learn how to check for normality.

Strategy

• If you have a large sample of, say, 30 or more measurements, then don’t worry about normality, and calculate a t-interval.

• If you have a small sample and your data are normally distributed, then calculate a t-interval.

• If you have a small sample and your data are not normally distributed, then stay tuned.