Question

A researcher wants to estimate the difference between the proportions of users of two shampoos who are satisfied
with the product. In a sample of 400 users of Fancy Pants Shampoo taken by this researcher, 78 said they are
satisfied. In another sample of 500 users of Smooth Operator Shampoo taken by the same researcher, 92 said they
were satisfied.
We are going to test if we can conclude that the proportion of Fancy Pants users who are satisfied is different from
the proportion of Smooth Operator users who are satisfied with the product.
a. Define the relevant parameters and state the hypotheses. [3pts]
b. Find the observed sample statistic, pooled proportion, and the test statistic. Use proper notation for the sample
statistic ( x or ˆp ), and clearly identify your test statistic as either a z-score or a t-score. [4pts]
c. Find the p-value of your test statistic. You are required to graph the test statistic on its corresponding distribution,
shade the appropriate area, and provide correct calculator documentation. Box final answer. [3pts]
d. Make a generic decision about 0 H . [2pts]
f. Make a conclusion in the context of the problem. [2pts]
g. Construct a 99% confidence interval for the difference between proportion of Fancy Pants users and Smooth Operator users who are satisfied with the product. [5 pts]
h. Provide an interpretation of your interval in the context of the data. [2pts]

Answer 1

allet. Plants shampoo Di= no. of Fancy Users nie 400 Xe no. of selected fancy planis shampoo usess x = 78 n2= no. of smooth operator Shampoo Weos n2z soo R2 = no of selected smooth operator Shampoo wers x2 = 92. Ô 78 Pi = 0,195 ni 400 P2 = 92 soo = P2 = 0.184

Testing the null & allemative hypothesis. Ha! Pi = P2 Hoi P = P2 b) pooled proporth, P- nihit mir ni+n2 40040.195 + 500x 0.184 400+500 190 goo Ip-0, 189 9 = -P = 1- 0.189 = 0.811 use I score It n= goo then Test statistie is, i - P2 pq (t + h) 0.195- 0.184 10.1898 0811X (400 soo 0.011 0.02625 2=0.4189

9 p-value = P(Z > 0.4189) IP-value= 0.6752 P-value -0.3376 pralue 2 0.3376 11 22-0. 4189 2=0,4189 d) fail to reject Ho. g Conclusion: We conclude that that the propostion of Pancy Pants whers who are satisfied is not different from the propouth of smooth operator users who are satisfied with the produch, 99% GI hor Pi- P is, (Å - P) & Zal2|pq (t + too) Q = 0.01 &12 - 0.005 24/2 = 2.5758 (0.195- 0.184) † 2.5758 x 0.189 x 0.8111 ✓ (hoo Soo

0.ol1 I 2.5758 7 0.02625 +0.0676 (-0.0566 0.0786 h) 99% c. I for the difference bem propor tion of fancy pants wers f smooth operchor wers who are satisfied with product is 60.05667 010788) 0.0566 <Pi-P220,0788