Question

a) The research firm suggests that they should do a hypothesis test for Eropa. Why might this be a reasonable approach to take? State and explain the appropriate null and alternative hypotheses. (COMPANY DETAILS GIVEN IN THE END OF THE QUESTIONS)

b) Describe what a type 1 error and what a type 2 error would be in the context of this problem. Describe the consequences of each of these types of errors.

c) Eropa remembers that when she took Statistics 101 (she has a BA in Psychology) for some reason a significance level of 5% was always used for hypothesis tests. Explain to Eropa why she might not want to automatically use 5%. What factors should she consider when choosing the level of significance?

d) Getting firms to respond to a survey is expensive and usually requires some inducement for the respondent. The firm gives Eropa a flat price of $15,000 for 50 respondents to the survey (and a report) and $150 for each additional respondent. You don’t have enough information to tell Eropa what sample size she should use but please describe for her what factors she should consider when deciding how big a sample to buy.

DETAILS OF THE COMPANY:

Eropa, an entrepreneur, created a company that matches firms (hotels, conference centres, etc.) that need to hire temporary staff for events (weddings, galas, conference dinners, etc.) to people looking for temporary work. Most hotels do not have enough permanent staff for these large, occasional events so they rely on temporary staff. There are temp agencies that connect these employers with bartenders, servers and others wanting to pick up extra shifts; in fact Eropa used to work for one of those agencies. Eropa has created an online platform (analogous to Airbnb or Uber) to match employers and employees and she is successfully disrupting temp agencies in eastern Canada. Eropa is considering expanding to Western Canada but it won’t be cheap (she figures she needs $1.5 million if she expands) so she wants to get a sense of market demand first. Eropa knows that there are about 10,000 firms that stage events in Western Canada that might need to hire temporary staff. She figures that if each firm, on average needs to fill at least 500 shifts a year with temporary staff there will be enough demand to justify entering the market. So, since there is no publically available data on current demand, she hires a market research firm to survey a sample of the 10,000 firms to estimate average demand for temporary staff (in shifts per year).

Answer 1

a)  

A hypothesis test is a reasonable approach to get information about the average number of shifts filled with temporary staff in a year by each firm.

Let the average number of shifts filled with temporary staff in a year by each firm be denoted by .

Then according to the requirement of Eropa;

• null hypothesis H₀: = 500
• alternative hypothesis H₁: > 500

Thus the hypothesis test is given as:-

H₀: = 500 vs H₁: > 500

b)

Type 1 error is made when H₀ is true but it is rejected for being false with respect to the sample obtained. In this context, type 1 error is made when average number of shifts filled with temporary staff in a year by each firm is actually less than 500, but it is assumed to be at least 500 with respect to the sample obtained by the market research firm. As a result, Eropa will proceed with the business expansion under the assumption that she will incur profit, but will actually suffer a loss.

Type 2 error is made when H₀ is false but it is accepted for being true with respect to the sample obtained. In this context, type 2 error is made when average number of shifts filled with temporary staff in a year by each firm is actually at least 500, but it is assumed to be less than 500 with respect to the sample obtained by the market research firm. As a result, Eropa will reject the idea of business expansion under the assumption that she will suffer a loss, and thus will miss a potential profit.

c)

The consequences of type 1 error are more severe in this context. As level of significance is the maximum probability of type 1 error, a 5% significance level should not be automatically chosen by Eropa. She should take into account the maximum probability of type 1 error that can be allowed according to the risk of suffering loss she is willing to take, and then set the level of significance accordingly.

d)

A sample is obtained from the 10000 firms to obtain information about . For this purpose, an estimate of is obtained from the sample. This estimate depends on the sample size, consequently the variance of the estimate is also a function of the sample size.

According to the demand of the market research firm, the cost of sampling also depends on the sample size.

Therefore to determine the optimum sample size, Eropa has two strategies:-

• She fixes the variance of the estimate and then selects the sample size for which the cost of sampling reduces to its minimum possible value.
• She fixes the cost of sampling and selects the sample size for which the variance of the estimate reduces to its minimum possible value.