Inference for the Difference Between Two Means Section 9.4 Inference for the Difference Between Two Means

To determine the difference between two population means, can we take ?

To determine the difference between two population means, can we take ? Usually not because we often do not know the true population means.

To estimate the size of the difference between the mean of one population and the mean of another population, we can use . . . .

To estimate the size of the difference between the mean of one population and the mean of another population, we can use a confidence interval for the difference between two means, .

To estimate the size of the difference between the mean of one population and the mean of another population, we can use a confidence interval for the difference between two means, . This is known as a two-sample t-interval.

statistic (critical value) (standard deviation of statistic) Confidence interval for the difference between two means has the standard form: statistic (critical value) (standard deviation of statistic)

Confidence interval for the difference between two means has the standard form: statistic (critical value) (standard deviation of statistic)

Confidence interval for the difference between two means has the standard form: statistic (critical value) (standard deviation of statistic)

Unlike the one-sample case, the sampling distribution of the statistic for the difference of two samples does not have a t-distribution.

Unlike the one-sample case, the sampling distribution of the statistic for the difference of two samples does not have a t-distribution. The exact distribution is not even known.

The exact distribution is not even known. However, it is known that the distribution is reasonably close to a t-distribution if

The exact distribution is not even known. However, it is known that the distribution is reasonably close to a t-distribution if the right number of degrees of freedom is used.

The exact distribution is not even known. However, it is known that the distribution is reasonably close to a t-distribution if the right number of degrees of freedom is used. So, how do you determine the right number of df?

For dealing with the difference between two means, df is approximated by a rather complicated rule.

For dealing with the difference between two means, df is approximated by a rather complicated rule.

What do you notice about the degrees of freedom (df) in this calculator display?

To construct a confidence interval for the difference between two means, what do we need to do?

To construct a confidence interval for the difference between two means, what do we need to do? Check conditions

To construct a confidence interval for the difference between two means , what do we need to do? Check conditions Do computations

To construct a confidence interval for the difference between two means , what do we need to do? Check conditions Do computations Give interpretation in context

Check Conditions 1) For survey,

Check Conditions 1) For survey, two samples randomly selected from two different populations?

Check Conditions 1) For survey, two samples randomly and independently selected from two different populations.

Check Conditions 1) For survey, two samples randomly and independently selected from two different populations. For experiment,

Check Conditions 1) For survey, two samples randomly and independently selected from two different populations. For experiment, two treatments randomly assigned to available experimental units.

Check Conditions 2) normality:

Check Conditions 2) normality: two samples must look like they came from normally distributed populations or

Check Conditions 2) normality: two samples must look like they came from normally distributed populations or Sample sizes are large enough that sampling distributions of sample means will be approximately normal

15/40 Guideline

Check Conditions 15/40 guideline can be applied to each sample or treatment group, although it is a bit conservative.

Check Conditions 15/40 guideline can be applied to each sample or treatment group, although it is a bit conservative. For difference of two means, we can allow our populations to be more skewed than we did previously for estimating mean of a population

Check Conditions 3) For survey,

Check Conditions 3) For survey, population sizes should be at least ten times larger than sample sizes for both samples.

Check Conditions 3) For survey, population sizes should be at least ten times larger than sample sizes for both samples. Remember, this condition does not apply to experiment.

Do Computations Confidence interval for difference between means of two populations, , is: where x1 and x2 are respective means of the two samples, s1 and s2 are the standard deviations, and n1 and n2 are the sample sizes.

Do Computations Because value of t* depends on complicated formula, use 2-SampTInt under STAT TESTS You can start with actual data or summary statistics.

Give Interpretation in Context and Link to Computations For survey 95% confidence interval: I’m 95% confident that if I knew the means of both populations, the difference between those means, , would lie in the confidence interval.

Give Interpretation in Context and Link to Computations For experiment: If all experimental units could have been assigned each treatment, I’m 95% confident that the difference between the means of the two treatment groups would lie in the confidence interval.

Give Interpretation in Context and Link to Computations In either case of survey or experiment, you must give the interpretation in context, describing the two populations or treatment groups.

You constructed a two-sample t-interval at the 95% confidence level for the difference between the mean test scores on a final stats exam for 4th hour and 6th hour classes. Your interval is (- 0.23, 4.51). Interpret your interval.

I’m 95% confident that the true difference between the mean test scores on a final exam for two stats classes, , is in the interval (- 0.23, 4.51).

I’m 95% confident that the true difference between the mean test scores on a final exam for two stats classes, , is in the interval (-0.23, 4.51). Because this interval contains 0, it is plausible that there is no difference in the true mean test scores for these two stats classes.

Page 631, P27 Remember, you must explain why you believe or do not believe the necessary conditions are met.

Page 631, P27 a) Problem states we can assume these volunteers are independent, random samples.

Page 631, P27 a) Problem states we can assume these volunteers are independent, random samples. Both dot plots are fairly symmetric with no outliers so it is reasonable to assume that both samples are taken from populations that are approximately normally distributed.

Page 631, P27 Both populations are more than 10 times their respective sample sizes (10 x 7 = 70 and 10 x 23 = 230) as there are more than 70 left-handed people and more than 230 right-handed people.

Page 631, P27 b) For consistency in our answer, use left-handed volunteers as sample 1 and right-handed volunteers as sample 2. Use 2-SampTInt.

To Pool or Not to Pool? Almost always select the unpooled option. The only situation in which the pooled procedure has definite advantage over unpooled is when the population standard deviations are equal but the sample sizes are unequal.

To Pool or Not to Pool? Almost always select the unpooled option. The only situation in which the pooled procedure has definite advantage over unpooled is when the population standard deviations are equal but the sample sizes are unequal. Population standard deviations usually unknown

Page 631, P27 2-SampTInt Inpt: Data Stats x1: 59.57 C-level: .95 sx1: 14.77 Pooled: No Yes n1: 7 Calculate x2: 58 sx2: 15.71 n2: 23

Page 631, P27 b) Left-handed volunteers are sample 1 and right-handed volunteers are sample 2. (-12.76, 15.899)

Page 631, P27 c) I’m 95% confident that the difference between the mean distance all left-handed volunteers could walk before crossing a sideline and the mean distance all right-handed volunteers could walk before crossing a sideline is in the interval (-12.76, 15.899).

Page 631, P27 d) Because the interval contains 0, we do not have statistically significant evidence that left- and right-handed volunteers differ in the mean number of yards they can walk before crossing a sideline.

Page 632, E53

Page 632, E53 a) The treatments, raised in long days or raised in short days, were randomly assigned to subjects.

Page 632, E53

Page 632, E53

Page 632, E53 a) Both distributions are moderately skewed, but neither has any outliers. Since the distribution of the difference reduces skewness, and the t-procedure is robust against non-normality, the conditions for inference are adequately met to proceed.

Page 632, E53 b. Interval?

Page 632, E53 b. (0.50251, 9.46)

Page 632, E53 c. The difference between the mean enzyme concentration of all eight hamsters had they all been raised in short days and the mean concentration had they all been raised in long days.

Page 632, E53 d. Because 0 is not in the confidence interval, and the treatments were randomly assigned, Kelly has statistically significant evidence that the difference in enzyme concentrations between the two groups of hamsters is due to the difference in the amount of daylight.

Questions?