Chpt 10 - Estimation and Hypothesis Testing

Question 1

When using two independent samples, do the sample sizes need to be equal?

Answer 1

no, they can be equal or unequal

Question 2

How do we know if 2 samples are independent of if we should use pairs?

We want to check whether a new drug increases memory performance, give an example of how we could set this up using an independent test and a paired test?

Answer 2

In an independent test, or a non-pooled t-test, the selection of sample 1 does not depend on the sample 2; it is more of a difference between two groups

A paired t-test is a one sample t-test on the difference of each pair of the two selected samples, the two samples are not independent because each observation in sample 1 is related to an observation in sample 2; it is often the a difference in a group between 2 points in time

Example: We want to check whether a new drug increases memory performance

Non-pooled t-test: You give 20 people the drug and 20 people the placebo

Paired t-test: You test the memory performance of 40 people before and after they take the medicine

Question 3

What are the different hypotheses used for a non-pooled t-test?

What are the critical values for each?

Answer 3

Two-Tailed
μ1 ≠ μ2
Ho: μ1 - μ2 = 0
Ha: μ1 - μ2 ≠ 0
C1: -tα/2
C2: tα/2

Left-Tailed
μ1 < μ2
Ho: μ1 - μ2 => 0
Ha: μ1 - μ2 < 0
C1: -tα/2

Right Tailed
μ1 > μ2
Ho: μ1 - μ2 =< 0
Ha: μ1 - μ2 > 0
C2: tα/2

Question 4

What is the test statistic for a non-pooled t-test

Answer 4

to =

√(s1 squared/n1)+(s2 squared/n2)

Question 5

What is the short way of determining the degrees of freedom for a non-pooled t-test?

What degree of freedom would we use if:
n1 = 35
n2 = 30

Answer 5

df = min (n1-1, n2-1)

= min (35-1, 30-1)

= min (34, 29)

So df = 29

Question 6

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Set up the hypotheses

Answer 6

Ho: μ1 -μ2 = 0

Ha: μ1 -μ2 ≠ 0

Question 7

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Check assumptions

Answer 7

Normal distribution - We don’t have any information about the parent population but the samples are 35 and 30, so they both meat the central limit theorem

Simple random sample ✓

Independently selected ✓

Question 8

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Find critical values

Answer 8

α = 5% = 0.05

α/2 = 0.025

degrees of freedom
= min(n1-1, n2-1)
= min (35-1, 30-1)
= 29

C1 = -tα/2 = -t0.025 = -2.045

C2 = tα/2 = t0.025 = 2.045

Question 9

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Calculate the test statistic

Answer 9

to =

√(s1 squared/n1)+(s2 squared/n2)

√(26.21squared/35)+(23.45squared/30)

= 2.4363

Question 10

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Compare

Test statistic: 2.4363
C1; -2.045
C2: 2.045

Answer 10

Since to is 2.4363, it is in the rejection region, therefore we reject Ho

Question 11

Determine whether the mean salaries of collage faculty in private and public institutions are different by using the two samples provided at the 5% significance level. All of the numbers in this example are in thousands of dollars

Assume that 35 faculty members from private institutions (sample 1) and 30 faculty members form public institutions (sample 2) are randomly and independently selected. Sample 1 has a mean of 88.19 and standard deviation of 26.21 . Sample 2 has a mean of 73.18 and a standard deviation of 23.45.

Interpret:

Since to is 2.4363, it is in the rejection region, therefore we reject Ho

Answer 11

At the 5% significance level, the data provides sufficient evidence to conclude that the mean salaries of college faculty in private and public institutions are different

Question 12

When using a p value approach, when do we reject Ho?

Answer 12

When the p value is less than the significance level

Question 13

How do we determine a confidence interval for 2 independent samples?

Answer 13

(x̄1-x̄2) ± tα/2√(s1squared/n1)+(s2squared/n2)

degrees of freedom = n-1

Question 14

What is the test statistic for paired t-tests?

Answer 14

to = x̄d / (Sd/√n)

x̄d = sample mean of the sample difference

Sd = sample standard deviation

degrees of freedom = number of pairs - 1

Question 15

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Set up the hypotheses

Answer 15

Ho: μ1 - μ2 = 0
Ha: μ1 - μ2 > 0

It can also be said as:

Ho: μd = 0
Ha: μd > 0

Question 16

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Check the assumptions

Answer 16

Normal distribution - the sample size of 10 is not large enough so we need to assume that the parent population is normally distributed

normal random sample

Question 17

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Find the critical values

Answer 17

α = 5% = 0.05

Degrees of freedom = n-1 = 10-1 = 9

tα = t0.05 = 1.833

Question 18

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Calculate the test statistics

Answer 18

x̄d = 36/10 = 3.6

to = x̄d / (Sd/√n)

= 3.6/(4.917/√10)

= 2.29

Question 19

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Compare

test statistic 2.29
C2 1.833

Answer 19

The test statistic is greater than the C2 level, so we should reject Ho

Question 20

Determine whether, in the US, the mean age of married men is older than the mean age of married women at the 5% significance level. Suppose that 10 married couples in the US are selected at random and that the ages, in years, were identified. The sum of the differences have been determined to be 36 and the sample standard deviation is 4.971.

Interpret

The test statistic is greater than the C2 level, so we should reject Ho

Answer 20

At the 5% significance level, the data provides sufficient evidence to conclude that the mean age of married men is older than the mean age of married women

Question 21

What is the formula for the confidence interval for paired samples?

Answer 21

x̄d ± tα/2*Sd/√n

degrees of freedom = n-1