Question

3. The Heldrich Center for Workforce Department found that 45% of Internet users received more than 10 email messages per day. Recently, a similar study on the use of email was reported. The purpose of the study was to see whether the use of email increased. a. Formulate the null and alterative hypotheses to determine whether an increase occurred in the proportion of Internet users receiving more than 10 email messages per day. b. If a sample of 420 Internet users found 208 receiving more than 10 email messages per day, what is the p-value? c. Using α =0.05, what is your conclusion? d. Apply the critical value method when  = 0.01.

Answer 1

The given problem is to test that there is an increase occurred in the proportion of Internet users receiving more than 10 email messages per day. Thus we use one sample proportion z test to solve this problem.

GIVEN:

The Heldrich center for Workforce Department found that 45% of Internet users received more than 10 email messages per day.

Sample size

Number of Internet users receiving more than 10 email messages per day

(a) HYPOTHESIS:

The hypothesis for one sample proportion z test is,

(That is, the proportion of Internet users receiving more than 10 email messages per day is 45%.)

(That is, the proportion of Internet users receiving more than 10 email messages per day is more than 45%.)

(b) TEST STATISTIC AND P VALUE CALCULATION:

The sample proportion of Internet users receiving more than 10 email messages per day is,

  

TEST STATISTIC:

The test statistic is,

which follows standard normal distribution

where p is the hypothesized value .

Thus

  

  

P VALUE:

The p value is,

{Since right tailed , we calculate }

From the z table, the probability value is the value with corresponding row 1.8 and column 0.05.

  

Thus the calculated p value is .

(c) CONCLUSION FOR P VALUE:

Given the significance level .

Since the calculated p value (0.0322) is less than the significance level , we reject null hypothesis and conclude that the proportion of Internet users receiving more than 10 email messages per day is more than 45%. In other words there is sufficient evidence to prove that there is an increase occurred in the proportion of Internet users receiving more than 10 email messages per day.

(d) CRITICAL VALUE METHOD:

Given the significance level .

The right tailed (since ) z critical value at significance level is .

DECISION RULE:

CONCLUSION:

Since the calculated z statistic value (1.85) is less than the z critical value (2.33), we fail to reject null hypothesis and conclude that the proportion of Internet users receiving more than 10 email messages per day is 45%. In other words there is no sufficient evidence to prove that there is an increase occurred in the proportion of Internet users receiving more than 10 email messages per day.