The Diadora annual report states that Diadora is one of the

The Diadora annual report states that Diadora is one of the largest sellers of athletic footwear in the world. Diadora\'s footwear products are primarily designed for athletic use, but also for casual and leisure wear. Historical data indicates that the average customer buys 2.1 pairs of sports shoes per year, with a population standard deviation of 3.9. If samples of 22 customers are taken

a) What is the standard error of the mean for the sample means?

b) What is the probability that the a given sample mean is between 3 and 6 pairs of shoes?

c) What is the probability that the difference between a given sample mean and the population mean is less than 0.18?

d) What is the probability a given sample mean is greater than 6 pairs of shoes?

Thank you!!

Solution

a)
Standard Error= sd/ Sqrt(n)
Where,
sd = Standard Deviation
n = Sample Size

Standard deviation( sd )=3.9
Sample Size(n)=22
Standard Error = ( 3.9/ Sqrt ( 22) )
= 0.831
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 3) = (3-2.1)/0.831
= 0.9/0.831 = 1.083
= P ( Z <1.083) From Standard Normal Table
= 0.8606
P(X < 6) = (6-2.1)/0.831
= 3.9/0.831 = 4.6931
= P ( Z <4.6931) From Standard Normal Table
= 1
P(3 < X < 6) = 1-0.8606 = 0.1394                  
d)
P(X > 6) = (6-2.1)/0.831
= 3.9/0.831 = 4.6931
= P ( Z >4.693) From Standard Normal Table
= 0