A study aims to understand the year-to-year changes in small business sales duri
ID: 3202763 • Letter: A
Question
A study aims to understand the year-to-year changes in small business sales during the winter holiday period in the US. Researchers randomly sampled 100 small businesses in December 2014 and found that 65 of them had an increase in sales relative to December 2013.
A) Find a 95% confidence interval for the proportion of businesses that had an increase in sales.
B) Find the sample size needed for the 99.7% confidence interval to be of the same width as the 95% confidence interval calculated above. Note that the 95% confidence interval calculated above was based on a sample of size 100.
C) Another research paper (funded by an agency known to be critical of the government’s policies) cited this study claiming that less than 2/3rd of the small businesses had an increase in sales. With what level of confidence can this statement be made?
D) Similar to the study done for December 2014, if a 95% confidence interval was computed for each of the months starting from January 2001 to December 2014 (that is 14x12 = 168 months), what is the probability that at least 95% of the confidence intervals cover the true proportions? Feel free to assume (and state clearly), if required, that the sales for each small business for each month are independent of that for any other combination of business and month. Clearly specify if you have used this assumption and how you have used it.
Explanation / Answer
Q1.
Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
No. of success(x)=65
Sample Size(n)=100
Sample proportion = x/n =0.65
Confidence Interval = [ 0.65 ±Z a/2 ( Sqrt ( 0.65*0.35) /100)]
= [ 0.65 - 1.96* Sqrt(0.002) , 0.65 + 1.96* Sqrt(0.002) ]
= [ 0.557,0.743]
Q2.
Compute Sample Size ( n ) = n=(Z/E)^2*p*(1-p)
Z a/2 at 0.003 is = 2.968
Sample Proportion = 0.65
ME = 0.09
n = ( 2.968 / 0.09 )^2 * 0.65*0.35
= 247.414 ~ 248
Q3.
Interpretations:
1) We are 95% sure that the interval [0.557 , 0.743 ] contains the true population proportion
2) If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion
Yes, it is true
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