t Test for One Sample

Why use a t test?The sampling distribution of t represents the

distribution that would be obtained if a value of t were calculated for each sample mean for all possible random samples of a given size from some population.

Just like the z sampling distribution!!

Why use a t test?

The figure indicates where the control and treatment group means are located. The question the t-test addresses is whether the means are statistically different.

Why use a t test?

When we are looking at the differences between scores for two groups, we have to judge the difference between their means relative to the spread or variability of their scores.

Gas Mileage Investigation (269)Hypothesis test summary for t test for a population mean

Research Problem

Does the mean gas mileage for some population of cars drop below the legally required minimum of 45 mpg?

Statistical Hypothesis

H0: µ ≥ 45

H1: µ < 45

Decision Rule

Reject H0 at the .01 level of significance if t≤ -3.365 (from Table B, Appendix C, given df = n-1 = 6 – 1 = 5).

Calculations

Given X = 43, sx = 0.89 (See table 13.1 on page 275 for computations)

t = 43 – 45 = -2.25 0.89

Decision

Retain H0 at .01 level of significance because t = -.2.25 is less negative than -3.365

Interpretation

The population mean gas mileage could equal the required 45 mpg or more. The manufacturer shouldn’t be penalized.

Table of critical values for tTable B of Appendix C page 520

Missing dfIf the desired number of degrees of freedom

doesn’t appear in the df column of Table B, use the row in the table with the next smallest number of degrees of freedom.

t ratio for a single population mean

t = sample mean – hypothesized population mean

estimated standard error

or X - µhyp

sx

Estimating the standard errorIf the population standard deviation is

unknown, it must be estimated from the sample.

sx = _s_ √n

Where s = SS √df

Sum of squares for X

SSx = Σ(X)2 – (ΣX)2

n

Calculations for a t test1. Define n2. Sum all X scores3. Find mean of X4. Square each X score5. Sum the squared Xs6. Calculate SS7. Calculate s8. Calculate sx

9. Assign value to µhyp

10. Calculate t

In classProgress Check 13.3

Confidence interval (p 276)

X ± (tconf)(sx)

Find value of tconf in Table B

HomeworkReview question 13.7, a and b, due Tuesday

4/7Show your work, answers in the back of the

book.