Your manager at Shape and Color Salon Headquarters recently stated in a meeting

Question

Your manager at Shape and Color Salon Headquarters recently stated in a meeting that 50 percent or more of the salon chain’s customers get color services in addition to haircuts. As such, she would like to do more marketing for the color services. However, the salon’s Director of Marketing is not sold on the idea and states that before making a change to the marketing strategy, this statement needs to be tested. You take a simple random sample of 180 customers. Of these, 97 customers respond that they received hair coloring services, as well as haircuts at the salon. Use this information to complete the following steps and determine if the marketing department should consider the change. a. Determine the population proportion. b. Calculate the sample proportion. c. Determine the mean and standard deviation of the sample distribution. d. Define the event of interest. e. If np and n(1-p) are both 5, convert to a standardized z-value. f. Use the Standard Normal Distribution Table to determine the probability for the event of interest.

Explanation / Answer

Given n = 180, x = 97

a) Your manager stated 50 percent or more of the salon chain’s customers get color services in addition to haircuts

i.e P = 0.50

H0: not 50 percent or more of the salon chain’s customers get color services in addition to haircuts

H1: 50 percent or more of the salon chain’s customers get color services in addition to haircuts

b) sample proportion p = x/n = 97/180 = 0.5389

c) Mean of sampling distributon of p is population proportion p = 0.50

SD of sampling distribution of p is Sqrt(PQ/n) = SQRT(0.50*0.50/180) = 0.03727

d) Here np = 180*0.5389 = 97 which is greater than 5

and nq = 180*(1-0.5389) = 82.998 which is greater than 5

The condition is satiesfied

e) Z = (p - P) / SE(p) = (0.5389 - 0.5) / 0.03727 = 1.04374

f) critical vlaue of Z = 1.96 at 5% los

Here Z value < Z critical value, we accept H0

Thus, we conclude that 50 percent or more of the salon chain’s customers get color services in addition to haircuts

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