Question

The management of a soda company wanted to determine the average number of ounces of soda consumed per resident in the state of Pennsylvania. Past trends indicate that the variation in soda consumption (σ) was 4 ounces. A 95% confidence level is required, and the error is not to exceed +/- 1/2 ounce. (a) What sample size would you recommend? Show your calculations. (b) The management wanted an estimate twice as precise as the initial precision level and an increase in the confidence interval to 99%. What sample size would you recommend? Show your calculations. Briefly comment on your results (in 2 to 3 sentences). 2. Clean Hair is a medium sized manufacturer of shampoo. The management of the company wanted to know how their sales compared with their main competitors: Silky Hair and Shiny Hair. A survey of 1,800 consumers indicated the following frequencies, with respect to their most recent shampoo purchased: Shampoo Number Buying Clean Hair 425 Silky Hair 1,175 Shiny Hair 200 Total: 1,800 Experience shows that 3 times as many households preferred Silky Hair to Clean Hair, while twice as many households preferred Clean Hair to Shiny Hair. The management wants to know from you if the historic tendency still holds. Please answer the question using the appropriate statistical tests.

Answer 1

Solution for Q1

Back-up Theory

Sample size, n, required to estimate population mean µ when σ is known is:

n = (σ²Z²_α/2)/E², where σ = 4 (given), Z_α/2 = upper (α/2)% point of N(0, 1) and E = desired level of accuracy (error).

Now, to work out the answer,

Part (a)

Given 95% confidence level, α/2 = 0.025 (2.5%) and hence Z_α/2 = 1.96 (from Standard Normal Tables). Also given is E = 0.5.

Substituting these values in the above formula,

n = {(4 x 1.96)/0.5}²

= 245.8624

Thus, required sample size is: 246 ANSWER

Part (b)

Given 99% confidence level, α/2 = 0.005 (0.5%) and hence Z_α/2 = 2.57583 (using Excel Function of N(0, 1). Also given is E = 0.25.

Substituting these values in the above formula,

n = {(4 x 2.57583)/0.25}²

= 1698.534

Thus, required sample size is: 1699 ANSWER 1

Comments

When confidence level increases Z value increases which in turn incases n. Further, doubling precision means halving E, and E being in the denominator, this also increases the sample sizes. Thus, there is double cumulative effect. The increase in n becomes profound since there is also a squaring factor. ANSWER 2