Testing a Proportion with Approximate Normality A large appa

Testing a Proportion with Approximate Normality A large apparel company sells clothes to both women and men. In a simple random sample of 180 customers, 95 are women. Let be the typical proportion of customers who are women. a) What can you say about type of the sampling distribution of the sample proportion? Explain. b) At the level of significance = 0.05, test H0: 0.54 versus H1: < 0.54. c) Find the pvalue for the test. Hint: Round off probabilities and values to 5 decimal places. Transform the sample proportion into its zscore. Determine probabilities based on some Excel probability and value lookups:

Solution

Set Up Hypothesis
Null, H0:P>=0.54
Alternate, H1: P<0.54
Test Statistic
No. Of Success chances Observed (x)=95
Number of objects in a sample provided(n)=180
No. Of Success Rate ( P )= x/n = 0.5278
Success Probability ( Po )=0.54
Failure Probability ( Qo) = 0.46
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.52778-0.54/(Sqrt(0.2484)/180)
Zo =-0.329
| Zo | =0.329
Critical Value
The Value of |Z | at LOS 0.05% is 1.64
We got |Zo| =0.329 & | Z | =1.64
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho
P-Value: Left Tail -Ha : ( P < -0.32901 ) = 0.37107
Hence Value of P0.05 < 0.37107,Here We Do not Reject Ho

[ANSWERS]
a. apparel company sells clothes to women more than Men
b. H0:P>=0.54, H1: P<0.54
c. Ha : ( P < -0.32901 ) = 0.37107