BA 2120 - LS 11

A pooled estimate for σ2 is denoted s2psp2 and is calculated as ______

(n1−1) s21 + (n2−1) s22n1 + n2 − 2

Suppose we wish to test H0: μ1 - μ2 = -10 versus Ha: μ1 - μ2< -10 by using a pooled variance t test. If the difference of the sample means is -13.8; n1= n2 = 35; and s1= 2.7 and s2= 3.1, what is the value of the test statistic and what do you decide when using α = 0.025?

-5.47; reject H0and decide the difference is less than -10

Given sample one has standard deviation 2 and sample size 20 while sample two has standard deviation 4 and sample size 30, what are the degrees of freedom for the unequal variances t statistic?

45 Reason: df = (420+1630)2(420)219+(1630)229420+16302420219+1630229 = 45.14; round down to 45

Suppose the first sample is of size 100 and has a variance of 24. Also suppose the second sample is of size 100 and has a variance of 6. Calculate the degrees of freedom for the unequal variances test statistic.

145

Suppose the first sample is of size 10 and has a variance of 14. Also suppose the second sample is of size 12 and has a variance of 16. What is the pooled estimate for a common population variance?

15.1

Suppose you are comparing two population means and the population variances are equal but unknown. Under what conditions is sp2 = (s12+s22)/2 ?

When n1= n2

Suppose you are comparing two population means and the population variances are equal but unknown. Under what conditions is sp2 = (s12+s222)s12+s222 ?

When n1= n2

Given the confidence interval [(x⎯⎯1−x⎯⎯2)±tα/2s2p(1n1+1n2)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√]x¯1-x¯2±tα/2sp21n1+1n2 for the mean difference between two normally distributed populations having equal variances, the t distribution is based on ______ degrees of freedom.

n1 + n2 - 2

The sampling distribution of the test statistic (x⎯⎯1−x⎯⎯2)−D0(s2pn1+s2pn2)x¯1-x¯2-D0sp2n1+sp2n2 is a t distribution having ______ degrees of freedom.

n1 + n2 - 2

Combining the results of two independent random samples to compute a single estimate of σ2 is called using a(n) ______ estimate of σ2.

pooled

The sampling distribution of ______ describes the probability distribution of all possible differences between two sample proportions.

p̂1 - p̂2

Fα denotes the point under the F distribution curve that gives a ______ -hand tail area equal to α

right

The estimate of the standard deviation of x⎯⎯1−x⎯⎯2x¯1-x¯2 is

s2p(1n1+1n2)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√

The test statistic t = (d−D)/(sd/√n) is used when ______.

testing the mean of a population of paired differences.

When constructing a confidence interval for the difference between proportions, the constraint n2(1 - p̂p̂2) ≥ 5 ensures __________.

the two samples are large enough

The formula [(x1−x2)x1-x2 ± tα/2 s21n1+s22n2⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√s12n1+s22n2] is used to construct a 100(1 - α)% confidence interval for μ1 - μ2 when two independent samples from two normally distributed populations having ______ variances.

unequal or different

If the test statistic used to compare two population means is x⎯⎯1−x⎯⎯2s2p(17+111)√x¯1-x¯2sp217+111 then the null hypothesis and the two sample sizes are

μ1 − μ2 = 0; 7 and 11

Suppose the first sample is of size 15 and has a variance of 8.6. Also suppose the second sample is of size 20 and has a variance of 10.2. What is the pooled estimate for a common population variance?

9.52 Reason: s2pp2 = (n1−1)s21+(n2−1)s22n1+n2−2(n1-1)s12+(n2-1)s22n1+n2-2 = (15−1)(8.6)+(20−1)(10.2)15+20−2(15-1)(8.6)+(20-1)(10.2)15+20-2 = 314.233314.233 = 9.52

Suppose we have taken independent random samples of sizes n1 = 7 and n2 = 7 from two normally distributed populations. Also suppose we obtain xx1 = 240, xx2 = 210, s1 = 5 and s2 = 6. Using the unequal variances procedure, construct a 95% confidence interval for μ1 - μ2.

30 ± 2.201 257+367⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√

Which of the following are examples of a paired difference experiment?

After drinking a sports drink, 20 people ran as far as possible in an hour. The same people repeated the running task after drinking water. 300 voters answered a questionnaire and then watched a video. Afterwards, the same voters answered a follow-up questionnaire.

In a random sample of 10 Sobeys grocery stores, researchers found that a standardized shopping list generated a sample mean of $121.916 and a sample standard deviation of $1.3982. In an independent random sample of 10 Metro grocery stores, the same list generated a sample mean of $114.807 and a sample standard deviation of $1.8403. Using the unequal variances procedure, a 90% confidence interval for the difference between the two grocery stores is closest to ______.

[$5.83, $8.39]

In a random sample of 10 Fresh Foods grocery stores, researchers found that a standardized shopping list generated a sample mean of $121.916 and a sample standard deviation of $1.3982. In an independent random sample of 10 Metro grocery stores, the same list generated a sample mean of $114.807 and a sample standard deviation of $1.8403. Using the equal variances procedure, a 90% confidence interval for the difference between the two grocery stores is closest to ______.

[$5.84, $8.38]

A pooled estimate for σ2 is denoted s2p and is calculated as ______

[(n1−1) s21 + (n2−1) s22] / [n1 + n2 − 2]

Given two independent samples from two normally distributed populations having unequal variances, what is the 100(1 - α)% confidence interval for μ1 - μ2?

[(x⎯⎯1−x⎯⎯2)±tα/2s21n1+s22n2⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√]

Assume we have selected two independent random samples and that p̂p̂1 = 800/1000 and p̂p̂2 = 950/1000. A 95% confidence interval for p1 - p2 is closest to ______.

[-0.178, -0.122]

A random sample of 400 television ads in the United Kingdom revealed that 142 use humor, while a random sample of 500 television ads in the United States revealed that 122 use humor. A 95% confidence interval for pUK - pUS is closest to ______.

[0.051, 0.171]

Suppose a sample of 49 paired differences that has been randomly selected from a normally distributed population of paired differences yields a sample mean of dd = 5 and a sample standard deviation of sd = 7. A 95% confidence interval for μd is closest to ______.

[3, 7]

Suppose we have taken independent random samples of sizes n1 = 7 and n2 = 7 from two normally distributed populations. Also suppose we obtain xx1 = 240, xx2 = 210, s1 = 5 and s2 = 6. Using the equal variances procedure, construct a 95% confidence interval for μ1 - μ2.

[30 ± 2.179 30.57+30.57⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√30.57+30.57]

Let d and sd be the mean and standard deviation of a sample of n paired differences that have been randomly selected from a normally distributed population. What is the 100(1 - α)% confidence interval for the mean of the population of paired differences μd?

[d ± tα/2 sd√n]

Given the confidence interval [d ± tα/2 sd√n] for the mean of a population of paired differences, the t distribution is based on ______ degrees of freedom.

n - 1

When testing the mean of a population of paired differences, the test statistic is a t distribution having ______ degrees of freedom.

n - 1

When constructing a confidence interval for the difference between proportions, the two samples must be large enough. This means that _____.

n2p̂2 ≥ 5 n1(1 - p̂1) ≥ 5 n2(1 - p̂2) ≥ 5 n1p̂1 ≥ 5

An experiment in which we have obtained two different measurements on the same n units is called a(n) ______ experiment.

paired difference

The sampling distribution of p̂1 - p̂2 describes the probability distribution of all possible differences between two _________.

sample proportions