Reading 11 Hypothesis Testing

Question 1

Test the hypothesis that the population correlation coefficient equals zero.

Answer 1

H0: ρ=0; HAlt: ρ≠0

t
Calc
=
r
√
n
−
2
√
1
−
r
2
 where degrees of freedom = 2

Reject H0 if tCalc is greater than t-critical from the t-table for the appropriate degrees of freedom and level of significance.

Question 2

Identify the rejection rules that apply when trying to determine whether a population mean is greater or less than the hypothesized value.

Answer 2

Greater than the hypothesized value:
Reject H0: Test statistic > positive critical value
Fail to reject H0: Test statistic ≤ positive critical value
Less than the hypothesized value:
Reject H0: Test statistic < negative critical value
Fail to reject H0: Test statistic ≥ negative critical value

Question 3

Give the formula for degrees of freedom and the pooled estimator of the common variance for a test concerning population means of two normally distributed populations with variances assumed equal.

Answer 3

df = n1 + n2 – 2

s
2
p
=
s
2
1
(
n
1
−
1
)
\+
s
2
2
(
n
2
−
1
)
n
1
\+
n
2
−
2

s
2
1
=
variance of the first sample

s
2
2
=
 variance of the second sample

Question 4

Explain the confidence interval and distinguish between the confidence interval and hypothesis test.

Answer 4

A confidence interval is the range of values within which a population parameter is estimated to lie: the acceptance region of the null hypothesis.

A hypothesis test is used to determine whether the hypothesized value of the population mean, μ0, lies within the confidence interval (1 − α) where we fail to reject the null, or within a rejection region (α) where we reject the null in favor of the alternate.

Question 5

Describe the chi-square test, give the equation, and list three important features of it.

Answer 5

A chi-square test involves comparing the calculated test statistic to the critical chi-square value for the level of significance and degrees of freedom.

χ
2
=
s
2
(
n
−
1
)
σ
2
0

n = sample size; s2 = sample variance; 
σ
2
0
= hypothesized value for population variance

Three important features are as follows:

It is asymmetrical.
It is bounded by zero. Chi-square values cannot be negative.
It approaches the normal distribution in shape as the degrees of freedom increase.

Question 6

How is the test statistic calculated?

Answer 6

Test statistic
=
Sample statistic
−
Hypothesized value
Standard error of the sample estimate
=
––
X
−
μ
0
s
/
√
n

Question 7

When is the z-test used, and what is the z-statistic compared to in a z-test?

Answer 7

To conduct hypothesis tests of the population mean when the population is normally distributed and its variance is known.

The z-statistic is compared to the critical z-value at the given level of significance.

Question 8

The t-test is used when the variance of the population is unknown and which 2 conditions hold?

Answer 8

The sample size is large.

The sample size is small, but the underlying population is normally distributed or approximately normally distributed.

Question 9

Describe the structure of hypothesis tests concerning the variance of two populations.

Answer 9

One-tailed tests:

H
0
\:
σ
2
1
≤
σ
2
2
;
H
a
\:
σ
2
1
>
σ
2
2
H
0
\:
σ
2
1
≥
σ
2
2
;
H
a
\:
σ
2
1
<
σ
2
2

Two-tailed tests: 
H
0
\:
σ
2
1
=
σ
2
2
;
H
a
\:
σ
2
1
≠
σ
2
2

σ
2
1
= variance of Population 1

σ
2
2
= variance of Population 2

Question 10

How are the hypotheses describing the tests of means of two populations structured?

Answer 10

H0: μ1 − μ2 = 0; Ha: μ1 − μ2 ≠ 0 to test if two populations’ means are not equal.

H0: μ1 − μ2 ≥ 0; Ha: μ1 − μ2 < 0 to test if Population 1 mean is less than Population 2.

H0: μ1 − μ2 ≤ 0; Ha: μ1 − μ2 > 0 to test if Population 1 mean exceeds Population 2.

Question 11

What is the result of decreasing the significance level?

Answer 11

Decreasing the significance level shrinks the rejection region and reduces the probability of a Type I error. This means reducing the power of the test and increasing the probability of failing to reject a false null hypothesis; i.e., increases P(Type II error).

Question 12

Explain hypothesis testing.

Answer 12

The process of evaluating the accuracy of a statement regarding a population parameter given sample information. A hypothesis is a statement about the value of a population parameter developed to test a theory.

Question 13

Mathematically state the t-value for hypothesis tests concerning the mean of a single population.

Answer 13

t
=
––
X
−
μ
0
s
/
√
n

––
x
= sample mean

μ0 = hypothesized population mean

s = standard deviation of the sample

n = sample size

Question 14

In the process of hypothesis testing, the decision whether to use critical values based on the z-distribution or the t-distribution depends on what?

Answer 14

Sample size

The distribution of the population

Whether the variance of the population is known

Question 15

Describe the alternate hypothesis and state it mathematically.

Answer 15

We are trying to validate the alternate hypothesis (Ha), the statement that will only be accepted if the sample data provides convincing evidence of its truth.

The alternate hypothesis: 
H
a
\:
μ
>
μ
0
H
a
\:
μ
<
μ
0
H
a
\:
μ
≠
μ
0

Question 16

What is the null hypothesis?

Answer 16

The null hypothesis (H0) generally represents the status quo and is the hypothesis that we are interested in rejecting. This hypothesis will not be rejected unless the sample data provides sufficient evidence to reject it.

It is stated as follows regarding the mean of the population:

H
0
\:
μ
≤
μ
0
H
0
\:
μ
≥
μ
0
H
0
\:
μ
=
μ
0

μ = population mean

μ0 = hypothesized value of the population mean

Question 17

Differentiate between Type I and Type II errors.

Answer 17

Type I error: Rejecting a correct null hypothesis.

Type II error: Failing to reject an incorrect null hypothesis.

Question 18

What information is necessary to determine a critical value?

Answer 18

Relevant distribution

Sample size

Level of significance

Question 19

Describe the p-value and describe the meaning of a p-value that is lower and higher than the required level of significance.

Answer 19

The p-value is the smallest level of significance at which the null hypothesis can be rejected. It represents the probability of obtaining a critical value that would lead to rejection of the null hypothesis.

If the p-value is lower than the required level of significance, we reject the null hypothesis.

If the p-value is greater than the required level of significance, we fail to reject the null hypothesis.

Question 20

Why are the results of parametric and non-parametric tests both presented?

Answer 20

Gauging sensitivity of conclusions to the assumptions underlying parametric tests. Parametric tests are preferred if their assumptions are met because they usually offer more crisp and reliable conclusions.