Question

A sample of 100 account balances of a credit company showed an average balance of $3,200 with a standard deviation of $316.

Formulate the hypotheses that can be used to determine whether the mean of all account balances is significantly different from $3000.

Conduct a full hypothesis test using the p-value approach. Let α = .05.

#2

During the recent primary elections, the democratic presidential candidate showed the following pre-election voter support in Alabama and Mississippi.
State
Voters Surveyed
Voters in favor of Democratic Candidate
Alabama
750
400
Mississippi
900
560

We want to determine whether or not the proportions of voters favoring the Democratic candidate were the same in both states. Conduct a full hypothesis test using the p-value approach at 92% confidence.

#3

In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the United States, the following information was gathered.
East
West
n = 37
N = 41
X = $72,500
X= $74,000
S = 3000
S= 2500

Develop an interval estimate for the difference between the average yearly incomes of the marketing managers in the East and West. Use α = 0.05.

At 95% confidence, use the p-value approach and conduct a full hypothesis test to determine if the average yearly income of marketing managers in the East is significantly different from the West.

Answer 1

1) A sample of 100 account balances of a credit company showed an average balance of $3,200 with a standard deviation of $316. Formulate the hypotheses that can be used to determine whether the mean of all account balances is significantly different from $3000. Conduct a full hypothesis test using the p-value approach. Let α = .05.

GIVEN:

Sample size

Sample average balance $

Sample standard deviation $

HYPOTHESIS:

$ (That is, the mean of all account balances is not significantly different from $3000.)

$ (That is, the mean of all account balances is significantly different from $3000.)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

which follows t distribution with degrees of freedom.

DEGREES OF FREEDOM:

CALCULATION:

P VALUE:

The two tailed p value for the given two tailed hypothesis test with 99 degrees of freedom is .

R CODE:

DECISION RULE:

CONCLUSION:

Since the calculated p value (7.1985 e-09) is less than the significance level , we reject null hypothesis and conclude that the mean of all account balances is significantly different from $3000.

2) During the recent primary elections, the democratic presidential candidate showed the following pre-election voter support in Alabama and Mississippi.
State Voters Surveyed Voters in favor of Democratic Candidate
Alabama 750 400
Mississippi 900 560

We want to determine whether or not the proportions of voters favoring the Democratic candidate were the same in both states. Conduct a full hypothesis test using the p-value approach at 92% confidence.

GIVEN:

Sample size of voters surveyed in Alabama

Sample size of voters surveyed in Mississippi

Number of Alabama voters in favor of Democratic Candidate

Number of Mississippi voters in favor of Democratic Candidate

HYPOTHESIS:

(That is, the proportions of voters favoring the Democratic candidate were the same in both states Alabama and Mississippi.)

(That is, the proportions of voters favoring the Democratic candidate were not the same in both states Alabama and Mississippi.)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

which follows standard normal distribution

where

is the sample proportion of Alabama voters in favor of Democratic Candidate.

is the sample proportion of Mississippi voters in favor of Democratic Candidate.

is the pooled sample proportion.

CALCULATION:

The sample proportion of Alabama voters in favor of Democratic Candidate is,

The sample proportion of Mississippi voters in favor of Democratic Candidate is,

The pooled sample proportion is,

  

  

Now

  

  

  

P VALUE:

The two tailed p value is,

  

From z table, the probability value is the value with corresponding row 3.6 and column 0.09.

Thus the calculated p value is .

DECISION RULE:

CONCLUSION:

Since the calculated p value (0.0002) is less than the significance level , we reject null hypothesis and conclude that the proportions of voters favoring the Democratic candidate were not the same in both states Alabama and Mississippi.

3) In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the United States, the following information was gathered.
East West
n = 37 N = 41
X = $72,500 X= $74,000
S = 3000 S= 2500

At 95% confidence, use the p-value approach and conduct a full hypothesis test to determine if the average yearly income of marketing managers in the East is significantly different from the West.

Develop an interval estimate for the difference between the average yearly incomes of the marketing managers in the East and West. Use α = 0.05.

GIVEN:

Sample size of marketing managers in the East

Sample size of marketing managers in the West

Sample average yearly incomes in East $

Sample average yearly incomes in West $

Sample standard deviation in East

Sample standard deviation in West

HYPOTHESIS:

(That is, the average yearly income of marketing managers in the East is not significantly different from the West.)

(That is, the average yearly income of marketing managers in the East is significantly different from the West.)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

which follows t distribution with degrees of freedom.

where the pooled standard deviation is,

DEGREES OF FREEDOM:

CALCULATION:

The pooled standard deviation is,

  

  

P VALUE:

The two tailed p value for the given two tailed hypothesis test with 76 degrees of freedom is .

R CODE:

DECISION RULE:

CONCLUSION:

Since the calculated p value (1.9816) is greater than the significance level , we fail to reject null hypothesis and conclude that the average yearly income of marketing managers in the East is not significantly different from the West.

INTERVAL ESTIMATE FOR DIFFERENCE BETWEEN TWO POPULATION MEANS:

The formula for 95% confidence interval for difference between two population means is,

where is the t critical value with degrees of freedom at significance level .

The t critical value with degrees of freedom at significance level is .

The 95% confidence interval for difference between the average yearly incomes of the marketing managers in the East and West is,

  

  

  

Thus the 95% confidence interval for difference between the average yearly incomes of the marketing managers in the East and West is .