Transcript Section 7.1
More Inferences About Means Student’s t distribution and sample standard deviation, s Reconsider inferences about m, the population mean When we make a CI or calculate the test statistic for hypotheses involving m we use s, the standard deviation of the population from which we’re sampling. – The standard deviation of the sample mean is s/sqrt(n). – However, we often don’t know s! Using s, the sample standard deviation, to estimate s If we don’t know s, we need to estimate it from the sample. We use s as an estimate of s. – s is discussed in Ch. 1 (see p.48). – 1 2 xi x s n 1 Increased Uncertainty We’d like to make inference about m, the unknown population mean. – We use the sample mean as an estimator of m. – Now, we also use s as an estimate of s. 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 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 s is unknown. xm t s n – This test statistic has a t distribution with n 1 degrees of freedom. – The margin of error, m, for a CI is mt * s n where t* is the appropriate value from the t distribution with n 1 degrees of freedom. Assumptions When we use the t distribution, we assume the population from which we’re sampling is normally distributed. 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 a = 0.10). 25 recent sales are randomly chosen and these have an average sales price of $148,000 and s = $62,000. Perform the t-test. – What assumptions are needed? – What hypothesis is supported? Example: Bottling Factory A factory fills 20 oz. bottles with soda. Assume the amount of soda in a bottle has a normal distribution. A random sample of bottles was taken from the factory line (data in P:\Data\Math\Radmacher\bottles.mtw). – Is there evidence (at a = 0.05) to make us think that the mean filling level is not 20 oz.? 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?