unit 9: conf intervals and hip tests for 2 means

Question 1

When is the Welch procedure a better choice? When is the pooled-variance t procedure a better choice?

Answer 1

• both assume normally dist
• poled variance assumed pop have equal variances (is variances very diff might want to use welch)
• effect of diff population variances is made worse if sample sizes are very different (pooled variance not appropriate)

Question 2

what is the meaning of

SE(X¯1 − X¯2)?

Answer 2

• SE(X¯1 − X¯2) is the estimate of the standard deviation of the sampling distribution of X¯1 − X¯2.
• (It estimates σX¯1−X¯2).
• It is a measure of the dispersion of X¯1 − X¯2 in repeated sampling.

Question 3

Would it make sense to test the hypothesis H0: X¯1 = X¯2? Why or why not?

Answer 3

No, this would not make sense. Statistical hypotheses always involve parameters and never statistics.

Question 4

A 95% confidence interval for µ1 − µ2 is found to be (2, 28).

(a) Give an example of a null hypothesis that would be rejected at α = 0.05.
(b) Give an example of a null hypothesis that would not be rejected at α = 0.05.

Answer 4

• any value outside the interval would be rejected at this con (ex 42)
• any value in the con interval not rejected (ex 20)

Question 5

When we use the pooled-variance t procedure, it is because we know the populations have the
same variance.

Answer 5

false We are assuming that the (unknown) population variances are equal.

Question 6

the pooled-variance t procedure works well, even when the population variances are a little
different. This is especially true if the sample sizes are similar

Answer 6

true

Question 7

It would be most appropriate to use the Welch procedure instead of the pooled-variance t
procedure if the sample variances are very different and the sample sizes are very different

Answer 7

true

Question 8

If the conclusions from the Welch procedure and the pooled-variance t procedure are very
similar, then it does not matter much which procedure is used.

Answer 8

true

Question 9

The Welch procedure is an exact procedure, as long as X¯1 = X¯2.

Answer 9

False. The Welch procedure

is an approximate procedure.

Question 10

If µ1 = µ2, the sampling distribution of X¯1 − X¯2 is approximately symmetric about 0 for large
sample sizes.

Answer 10

True. The sampling distribution of X¯1 − X¯2 has a mean of µ1 − µ2, and is approximately normal for large sample sizes.

Question 11

If X¯1 = X¯2, the sampling distribution of µ1 − µ2 is approximately symmetric about 0 for large sample sizes

Answer 11

False. µ1 − µ2 is a fixed quantity, and does not have a sampling distribution.

Question 12

SE(X¯1 − X¯2) is the true standard deviation of the sampling distribution of X¯1 − X¯2)

Answer 12

False. It is the estimate of this quantity.

Question 13

The pooled-variance t procedures work well, even when the variances for the two populations are
very different, as long as the sample sizes are very different as well.

Answer 13

False. The pooled-variance

procedure performs poorly in this situation.

Question 14

Suppose we are interested in testing the null hypothesis µ1 = µ2, against a two-sided alternative.
All else being equal, the greater the difference between µ1 and µ2, the greater the power of the
tes

Answer 14

True. The greater the difference, the easier it is to detect that difference

Question 15

Suppose we are constructing a confidence interval for µ1 − µ2. All else being equal, the greater
the difference between µ1 and µ2, the wider the interval.

Answer 15

False. The width does not depend on

µ1 − µ2.

Question 16

Suppose we are constructing a confidence interval for µ1 − µ2. All else being equal, the greater
the difference between X¯1 and X¯2, the wider the interval.

Answer 16

False. The width does not depend onX¯1 − X¯2.

X¯1 − X¯2 is the midpoint of the interval.

Question 17

Suppose we are constructing a confidence interval for µ1 − µ2. All else being equal, the greater
the sample sizes, the narrower the interval

Answer 17

true

Question 18

Suppose we wish to test H0: µ1 = µ2. We obtain random samples from the respective populations, run the appropriate test, and find that the p-value is 0.00000032. We can be very
confident that our results have important practical implications.

Answer 18

False. There is strong evidence of a difference in population means, but whether this is important in a practical sense is
an entirely different question

Question 19

If we test H0: µ1 = µ2 against a two-sided alternative and find a p-value of 0.32, then we know
that µ1 = µ2.

Answer 19

False. We do not have any evidence of a difference in population means, but we
do not know the population means, and we do not know if they are equal.

Question 20

The pooled sample variance used in a pooled-variance t procedure is a weighted average of the sample variances, and tends to be closer to the sample variance with the higher number of observations.

Answer 20

true

Question 21

The pooled-variance t procedure requires that the two populations be normally distributed, however because the Welch procedure is only an approximate procedure, it does not require this assumption.

Answer 21

false

Question 22

When applying the paired difference procedure, it is assumed that the differences between pairs of observations constitute a simple random sample from the population of differences.

Answer 22

true

Question 23

The pooled-variance t procedure is most appropriate when the observations between the two groups are dependent.

Answer 23

false

Question 24

If the sample sizes are similar, the pooled-variance t procedure will still work relatively well, even if the population variances are not quite the same.

Answer 24

true