More Inferences About Means Student’s t distribution and sample standard deviation, s

Reconsider inferences about , the population mean When we make a CI or calculate the test statistic for hypotheses involving  we use , the standard deviation of the population from which we’re sampling. –The standard deviation of the sample mean is  /sqrt(n). –However, we often don’t know  !

Using s, the sample standard deviation, to estimate  If we don’t know , we need to estimate it from the sample. We use s as an estimate of . – s is discussed in Ch. 1 (see p.48). –

Increased Uncertainty We’d like to make inference about , the unknown population mean. –We use the sample mean as an estimator of . –Now, we also use s as an estimate of . n This results in increased uncertainty about the sample mean we’re likely to obtain. –What distribution describes this uncertainty? –Student’s t distribution.

Student’s t distribution n The t distribution… –Is similar to the normal distribution. –Has heavier tails than the normal distribution. –Exists with varying degrees of freedom (d.f.). When degrees of freedom are low, tails are heaviest. As degrees of freedom increase without bound, the t distribution converges to the normal distribution.

T statistic The t-statistic, t, is used for inference of the mean of a population, when  is unknown. –This test statistic has a t distribution with n  1 degrees of freedom. –The margin of error, m, for a CI is where t * is the appropriate value from the t distribution with n  1 degrees of freedom.

Assumptions n When we use the t distribution, we assume the population from which we’re sampling is normally distributed. n However, hypothesis tests and CIs using the t distribution are “robust” inference techniques. –They can often be used for even very non-normal populations if n  40. –If n <15, we must be sure that population distribution is very close to normal.

Example: Housing Prices A real estate agency in a big city wants to test whether the mean home price exceeds $132,000 (using  = 0.10). n 25 recent sales are randomly chosen and these have an average sales price of $148,000 and s = $62,000. n Perform the t-test. –What assumptions are needed? –What hypothesis is supported?

Example: Bottling Factory n A factory fills 20 oz. bottles with soda. Assume the amount of soda in a bottle has a normal distribution. n A random sample of bottles was taken from the factory line (data in P:\Data\Math\Radmacher\bottles.mtw). –Is there evidence (at  = 0.05) to make us think that the mean filling level is not 20 oz.?

n You want to rent an unfurnished one bedroom apartment. You take a random sample of 10 apartments advertised in the Mount Vernon News and record the rental rates. Here are the rents (in $ per month): 500, 650, 600, 505, 450, 550, 515, 495, 650, 395 –Find a 95% CI for the mean monthly rent for unfurnished one bedroom apartments in the community. –Do these data give good reason to believe that the mean rent of all such apartments is greater than $500 per month?