1. Catherine Chao, Director of Marketing Research, is evaluating consumer acceptance of a new toothpaste package. She plans to use a 80% confidence interval estimate of the proportion of households which prefer the new packages; she will accept a 0.05 error. Previous studies indicate that new packaging has an approximately 70% acceptance rate. The sample size should be at least _______.
323
27
427
59

Answer: The sample size is 138
(Please check the values given in the problem; there appears to be an error as the answer is not matching any of the answers given)

No of tails= 2
This is a 2 tailed test
as we are checking the accuracy of plus minus 5% or 0.05

confidence interval= 80%
Z corresponding to 80% confidence interval and 2 tailed test is 1.2816

We have to see that Z* sp < 5%
or sp < 5.%/Z or 5.% / 1.2816
or sp < 0.039014 =5%/1.2816
But
sp=standard error of proportion=square root of (pq/n)
or n=(pq)/(sp^2)
p= 70%
q= 30%
pq= 0.21 =0.7 x 0.3
or n=(pq)/(sp^2)= 137.97 =0.21/0.039014^2

Therefore sample size required for plus minus 5% accuracy is 138 (rounding off)

check:

80% Confidence limits for proportions=
p= 70.00%
q=1-p= 30.00%
n=sample size= 138
sp=standard error of proportion=square root of (pq/n)= 3.90% =square root of ( 70.% * 30.% / 138)
Confidence level= 80%
Significance level=alpha (a) = 20% =100% -80%
No of tails= 2
This is 2 tailed because we are calculating upper and lower confidence level

Since sample size= 138 >= 30 use normal distribution
Z at the 0.2 level of significance 2 tailed test = 1.2816

Upper confidence limit= p+z*sp= 75.00% =70.%+1.2816*3.9%
Lower confidence limit= p-z*sp= 65.00% =70.%-1.2816*3.9%

80% Confidence limit:
Upper limit= 75.00%
Lower limit= 65.00%

2. Financial analyst Larry Potts needs a sample of 100 securities listed on either the New York Stock Exchange (NYSE) or the American Stock Exchange (AMEX). According to the Wall Street Journal's "Stock Market Data Bank," 2,531 NYSE securities and AMEX 746 securities were traded on the previous business day. Larry directs his staff to randomly select 77 NYSE and 23 AMEX securities. His sample is a ____________.

disproportionate systematic sample
proportionate stratified sample
disproportionate stratified sample
proportionate systematic sample

Answer: proportionate stratified sample

2531+746= 3277
2531/3277= 0.77
746/3277= 0.23
Therefore, proportionate, stratified sample (startified because NYSE and AMEX separated)

3. Which types of data are normally used with parametric statistics?
interval and ordinal
ratio and ordinal
nominal and ordinal
interval and ratio

Answer: interval and ratio