Question 1
Which one of the following statements is true to perform a z-test for the difference between two population mean ¦Ì1 and ¦Ì2?
a. Three conditions are required to perform such a test.
b. The samples must be independent and selected randomly.
c. Sample size must be at least 30.
d. All of the above.

The z-test tests if there is a difference between the means of two samples or between the means of one sample and a population. For a two-sample z-test, the samples both have to be simple random samples, and either the samples have to come from a normal population or they have to be large
(at least 30). You also have to know the standard deviations of the populations the samples come from.

I think the answer is b, but I'm not 100% sure, because the question is worded poorly. I know that b is definitely correct. I think c is wrong, because the sample size doesn't have to be over 30 if the samples come from a normal population. If you don't know if the population is normal or not, then c would be true. I'm not sure if a is right or not ... in the paragraph above, I gave you three conditions that need to be met, but I've never heard any statistics teacher actually say "three conditions are required."

Question 2
Two samples are independent if
a. Both of the samples are selected from same population.
b. One sample selected from a population is related to the sample selected from the same population.
c. One sample selected from a population is not related to the sample selected from the same population.
d. One sample selected from a population is not related to the sample selected from another population.

Can you have more than one right answer for these questions? The definition of independent samples are samples selected from the same population, or different populations, that have no effect on one another (i.e. they are not correlated). Usually the samples are from different populations, because, for example, you are using the samples to test if the means of the populations are statistically significantly different.

Question 3
Answer questions 3 and 4 from the following given data.
The data given below are taken from two independent samples collected randomly.
Claim: ¦Ì1 > ¦Ì2, ¦Á = 0.10, sample statistics:
x1 (bar) = 500, s1 = 30, n1 = 100
x2 (bar) = 510, s2= 30, n2 = 75
What is the value of test statistics?
a. 0
b. -10
c. -25
d. 25

Here, you want to use a two sample z-test (you use a z-test because you know the sample standard deviations). It is one-sided because you are testing whether or not one mean is larger than the other, instead of testing if they are equal or not (in which case you would use a two-sided test).

Usually, "test statistic" means the observed value of the z-statistic (here, it's z = -2.18; see below). In this case, since the hypothesis we're testing is ¦Ì1 > ¦Ì2, or, ¦Ì1 - ¦Ì2 > 0, the test statistic would be

¦Ì1 - ¦Ì2 = 500 - 510 = -10

The z-statistic is calculated as:

z = 500 - 510 = -10/4.58 = -2.1822
¡Ì(302/100 + 302/75)

Question 4
What is the value of standardized test statistics?
a. 4.33
b. -4.33
c. 2.1822
d. -2.1822 This is correct (see above)

Question 5
When sample sizes are less than 30 and the population variances are equal, we use t-test with standard error
a. ¦ÒX1-X2 = ¦Ò(cap) ¡Ì(1/n1 + 1/n2) and d.f. = n1 + n2 - 2

b. ¦ÒX1-X2 = ¡Ì(s12 /n1 + s22 /n2) and d.f. = smaller of n1 -1 or n2 - 1

C. Both A & B

d. Neither A nor B

The formula is ¦ÒX1-X2 = ¡Ì(s12 /n1 + s22 /n2) with d.f. = n1 + n2 - 2. Therefore, the answer is neither.

Question 6
Answer questions 6 & 7 from the following given data:
Ho: ¦Ì1 ¡Ü ¦Ì2, ¦Á = 0.01, sample statistics:
x1 (bar) = 45, s1 = 4.8, n1 = 16
x2 (bar) = 50, s2= 1.2, n2 = 14
Assume: ¦Ò1
2 ¡Ù ¦Ò2
2
What is the test statistics?
a. -5
b. 5
c. 2
d. -2
Question 7
What is the standardized test statistics?
a. -2.65
b. -4.025
c. 1.32
d. 3.067

We're going to use a 2 sample, one-sided t-test, not assuming equal variances. It's 2 sample because we're comparing 2 means, one-sided because we're testing whether one is larger than the other, and not assuming equal variances, because the problem said not to.

As in question 3, "test statistic" means standardized test statistic, but here it is 45 - 50 = -5.

The t-statistic is calculated as:

t = 45 - 50 = -5/1.242 = -4.025 (so the answer is b)
¡Ì(4.82/16 + 1.22/14)

Question 8
The requirement(s) to perform two-sample hypothesis test with dependent samples is/are:
a. Samples are selected randomly.
b. Samples are dependent.
c. Populations are normally distributed.
d. All of the above.

Just like the other tests mentioned, the populations have to be normally distributed (so c is true). The samples are dependent, which is why you are using a test with dependent samples (so b is true).

In such a hypothesis test, usually one sample is randomly selected, then an experiment or treatment is performed, and the values after the treatment are used as the second sample. So, the sample is originally randomly selected, but the second sample is dependent on the first. I would still say that a is true.

Therefore the answer is d.

Question 9
Answer questions 9 - 13 for a = 0.10 using the following problem statement:
To test whether a fuel additive improves gas mileage, investigators measured the gas mileage (in miles per gallon) of nine cars with and without the fuel additive. At a level of ¦Á = 0.10, can it be concluded that the additive improved gas mileage ? The results are shown below.
Identify the claim and Ho & Ha for this investigation.
Car 1 2 3 4 5 6 7 8 9
Gas mileage without additive 34.5 36.7 34.4 39.8 33.6 35.4 38.4 35.3 37.9
Gas mileage with fuel additive 36.4 38.8 36.1 40.1 34.7 38.3 40.2 37.2 38.7
a. Ho: ¦Ìd = 0 (claim); Ha : ¦Ìd ¡Ù 0
b. Ho: ¦Ìd ¡Ý 0 ; Ha : ¦Ìd < 0 (claim)
c. Ho: ¦Ìd ¡Ü 0 (claim); Ha : ¦Ìd > 0
d. Ho: ¦Ìd ¡Ý 0 (claim); Ha : ¦Ìd < 0

By the way, this is an example of a hypothesis test with dependent samples, mentioned in the last problem.

Since you are testing that the additive improved mileage, the alternative hypothesis should be that the mean of the group with the additive is greater than the mean of the group without the additive. Therefore the difference will be less than 0 (because we're subtracting the values with the additive from the values without the additive. The "claim" is always the alternative hypothesis.

b is true.

Question 10
Which one is the critical value to for this problem?
a. 1.611
b. 1.397
c. -1.611
d. -1.397

If you look at a t-distribution table with 8 degrees of freedom, you will see that the critical value for a = 0.10 is 1.397. This is for a one-sided test. Because we expect out t-value to be negative based on our claim that the difference is less than 0, the critical value is -1.397, and the answer is d.

Question 11
What is the value of d(bar)?
a. 1.611
b. 1.397
c. -1.611
d. -1.397

The mean difference is -1.6111 (subtract each pair to get 9 differences, then take the average of those numbers). The answer is c.

Question 12
What is the value of sd?
a. 0.56
b. 0.66
c. 0.77
d. 0.89

The standard deviation of the differences is 0.7705, so the answer is c.

Question 13
If we use t-test, what will be the standardized test statistic t?
a. -6.273
b. -3.653
c. -7.06
d. -3.0

The t-statistic is calculated as:

Here, d-bar is -1.611, the standard deviation is 0.77, and n - 9. So, t = -6.273, and has a one tailed p-value of 0.00012, so the answer is a.

Question 14 5
What will be the concluding decision for the following investigation?
Claim: p1 < p2; ¦Á = 0.1
x1 = 471, n1 = 942; x2 = 372, n2 = 620
A. left-tailed test, reject Ho
B. right-tailed test, reject Ho
C. left-tailed test, fail to reject Ho
D. right-tailed test, fail to reject Ho

Here, you would use a one-sided z-test (use z for proportions). The first group has a proportion of 0.4995 and the second group has one of 0.60. The first group has a smaller proportion than the second, so it would be a left handed test, and in fact, you would reject the null hypothesis at the 0.01 level. The answer is a.

Question 15
Which of the following is true as a condition necessary to use the chi-square goodness of fit test?
a. The observed frequencies must be obtained using a random sample.
b. Each expected frequency must be at least 5.
c. None of the above.
d. Both of A & B. d is correct.

Question 16
What is the expected frequency for n = 500, pi = 0.9?
a. 0.45
b. 4.50
c. 45.00
d. 450.00

It would be (0.9)(500) = 450.

Question 17
Answer questions 17 & 18 from the following table:
What is the expected frequency for 2nd row ? 2nd column, E2,2?
_ Ratings _
Size of restaurant Excellent Fair Poor Total
Seats 100 or fewer 182 203 165 550
Seats over 100 180 311 159 650
_ 362 514 324 1200
a. 165.92
b. 235.58
c. 196.08
d. 278.42

E F P T
less than 100 182 203 165 550
more than 100 180 311 159 650
total 362 514 324 1200

You calculate the expected frequency by multiplying the row and column totals (in bold) and dividing by the grand total (1200).

(650)(514)/1200 = 278.41667

The answer is d.

Question 18
What is the expected frequency for the 1st row - 3rd column, E1,3?
a. 148.50
b. 175.50
c. 278.42
d. 235.58

This is the same as the last problem:

(550)(324)/1200 = 148.5

Question 19
Which statement is not true for an F-distribution?
a. The F-distribution is a family of curves. true
b. The total area under each curve of an F-distribution is equal to 1. true
c. F-distributions are normally distributed. false - they have an F-distribution!
d. F-values are always at least 0. true

Question 20
Which one is the critical F-value for a two-tailed test when ¦Á = 0.1 and d.f.N = 4 and
d.f.D = 8 ?
a. 2.81
b. 2.69
c. 3.84
d. 3.69
Look at an F-distribution table for a = 0.1. For df = 4, 8, the critical value is 2.806. The answer is a.