Question

Last year, 45% of business owners gave a holiday gift to their employees. A survey of business owners conducted this year indicates that 35% plan to provide a holiday gift to their employees. Suppose the survey results are based on a sample of 80 business owners.

(A) How many business owners in the survey plan to provide a holiday gift to their employees this year?

(B) Suppose the business owners in the sample did as they plan. Compute the p-value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year.

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to four decimal places.)

p-value =

(C) Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts decreased?

Reject H₀. There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

Do not reject H₀. There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

Reject H₀. There is insufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

Do not reject H₀. There is insufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

(E) What is the smallest level of significance for which you could draw such a conclusion? (Round your answer to four decimal places.)

Answer 1

(A) How many business owners in the survey plan to provide a holiday gift to their employees this year?

About 80 business owners in the survey plan to provide a holiday gift to their employees this year.

(B) Suppose the business owners in the sample did as they plan. Compute the p-value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year.

Here, we have to use z test for the population proportion.

Null hypothesis: H₀: the proportion of business owners providing holiday gifts has not decreased from last year.

Alternative hypothesis: H_a: the proportion of business owners providing holiday gifts has decreased from last year.

H₀: p = 0.45 versus H_a: p < 0.45

The test statistic formula is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

n = sample size = 80

p̂ = x/n = 0.35

p = 0.45

q = 1 - p = 1 – 0.45 = 0.55

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.35 – 0.45)/sqrt(0.45*0.55/80)

Z = -1.7979

Test statistic = Z = -1.80

P-value = 0.0359

(by using z-table)

(C) Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts decreased?

We are given

Level of significance = α = 0.05

P-value = 0.0359

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

Answer: Reject H₀. There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year.

(E) What is the smallest level of significance for which you could draw such a conclusion?

Required Smallest level of significance = 0.0359