INFERENCE ABOUT MEANS

Chapter 23

CLT!!

If our data come from a simple random sample (SRS) and the sample size is sufficiently large, then we know the sampling distribution of the sample means is approximately normal with mean µ and standard deviation .

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Problem

If σ is unknown, then we cannot calculate the standard deviation for the sampling model.

We must estimate the value of σ in order to use the methods of inference that we have learned.

Solution

We will use s (the standard deviation of the sample) to estimate σ.

Then the standard error of the sample mean is

In order to standardize , we subtract its mean and divide by the standard deviation.

__________has the normal distribution N(0,1).

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s

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Problem

If we replace σ with s, then the statistic has more variation and no longer has a normal distribution so we cannot call it z. It has a new distribution called the t distribution.

t is the standard value. Like z, t tells us how many standardized units is from the mean µ.

When we describe a t-distribution we must identify its degrees of freedom because there is a different statistic for each sample size. The degrees of freedom for the one-sample t statistic is n – 1.

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The t distribution is symmetric about zero and is bell-shaped, but there is more variation so the spread is greater.

As the degrees of freedom increase, the t distribution gets closer to the Normal distribution, since s gets closer to σ.

We can construct a confidence interval using the t distribution in the same way we constructed confidence intervals for the z distribution.

Remember, the t Table uses the area to the right of t*.

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One sample t procedures are exactly correct only when the population is Normal. It must be reasonable to assume that the population is approximately normal in order to justify the use of t procedures.

When to use t procedures:

If the sample size is less than 15, only use t procedures if the data are close to Normal.

If the sample size is at least 15 but less than 40 only use t procedures if the data is unimodal and reasonably symmetric.

If the sample size is at least 40, you may use t procedures, even if the data is skewed.

Example

A coffee vending machine dispenses coffee into a paper cup. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is shortchanging the customer?

9.9 9.7 10.0 10.1 9.9 9.6 9.8 9.8 10.0 9.5

9.7 10.1 9.9 9.6 10.2 9.8 10.0 9.9 9.5 9.9

PHANTOMS!!

Example 2

A company has set a goal of developing a battery that lasts five hours (300 minutes) in continuous use. In a first test of these batteries, the following lifespans (in minutes) were measured: 321, 295, 332, 351, 311, 253, 270, 326, 311, and 288.

Find a 90% confidence interval for the mean lifespan of this type of battery.

PANIC!!!

If we wish to conduct another trial, how many batteries must we test to be 95% sure of estimating the mean lifespan to within 15 minutes?

To within 5 minutes?