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Section 7-2: Confidence Intervals for the Mean When σ Is Unknown

When one of the following is true, the confidence interval for the mean can be found by using the standard normal (Z) distribution in Section 7-1:

is known and the population is normally distributed or

the sample size is at least 30. When

is unknown, we can no longer use the standard normal distribution, but instead we will use a t distribution.

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Characteristics of the t-distribution The t distribution is similar to the standard normal distribution in these ways:

It is bell-shaped It is symmetric about the mean The mean, median, and mode are equal to 0 and are

located at the center of the distribution The curve never touches the x-axis.

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The t distribution differs from the standard normal distribution in the following ways:

The variance is greater than 1 (i.e., the t distribution has more variability than a standard normal distribution).

The t distribution is actually a family of curves (see pg. 786) based on the concept of degrees of freedom, which is related to sample size.

The shape of the t curve changes depending on the degrees of freedom.

As the sample size increases, the t distribution becomes more and more liked the standard normal distribution.

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Degrees of freedom (df) - the number of values that are free to vary after a sample statistic has been computed.

Example: Suppose the mean of 5 values if 10. What is

x ?

How many values can vary freely? We could vary 4 values of the 5. The degrees of freedom for a confidence interval for the mean

when is unknown is found by Sample size – 1 or n-1

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Formula for a Specific Confidence Interval for the Mean When σ is Unknown

The formula for a confidence interval about the mean when

is unknown is given by

X t 2

s

n

,

where

t /2 is the

1 /2 quantile of a t distribution with

n1 degrees of freedom. We will use Table F to find our t multiplier,

t /2.

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ASSUMPTIONS:

1. Our sample is a random sample from some population.

2. The population from which the sample came is normally distributed or the sample size n30.

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Example: Find the t multiplier (

t 2) for 98% confidence and 24

degrees of freedom.

1-=0.98 so we need =0.02 t/2 = 2.492

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Example: Find the t-multiplier (

t 2) for 90% confidence and

17 degrees of freedom. Note: Why do we need to use the t distribution instead of the Z distribution? Since we are estimating the population standard deviation

, we need to construct a wider confidence interval to take into account the uncertainty in our estimate of

. The t distribution changes the multiplier to the proper number to get the correct confidence level.

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Example: A sample of six college wrestlers had an average weight of 276 pounds with a sample standard deviation of 12 pounds. Find the 95% confidence interval of the true mean weight of all college wrestlers. If a coach claimed that the average weight of the wrestlers on his team was 310, would the claim be believable?

Assumptions: We have a random sample of college wrestlers. Since our sample size is under 30, we must assume that the average weight is normally distributed.

X 276

s 12

n 6

t /2 2.571

df 6-1 = 5

/2

12276 2.571 276 12.59528 (263,289)

6

sX t

n

Interpretation: We are 95% confident that the average weight of a college wrestler is between 263lbs and 289lbs.

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Example: The daily salaries of substitute teachers for eight local school districts is shown. What is the sample mean? Find the 99% confidence interval of the mean for the salaries of substitute teachers in the region.

60 56 60 55 70 55 60 55

Assumptions:

X

s

n

t /2

df

X t /2s

n

Interpretation:

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Constructing confidence intervals on a TI-83/84 for the Mean When σ is Unknown:

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Entering data on a TI-83/84:

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When do you use the z-table and when do you use the t-table?!?!

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