Help finding the sample size! Statistics

Question

A paint manufacturer uses a machine to fill gallon cans with paint.

 

a)The manufacturers wants to estimate the mean volume of paint the machine is putting in the cans within 0.25 ounces. determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.85 ounces.

 

b)Repeat part (a) using an error tolerance of 0.15 ounces.

 

c) Which error tolerance requires a larger sample size? Explain

Answer 1

A 90% confidence interval for μ is given by 

 

x +- (z^*)*σ/sqrt(n)  where  

 

x = x bar = sample mean

z^* = critical z-value for 90% = 1.645 from Z-table since P(Z<-1.645) = 0.05 and confidence interval is two sided 

σ = population standard deviation = 0.85 ounces 

n = number of samples 

 

You want the margin of error (this is the part of the confidence interval after the +- sign) to be within 0.25 ounces, so 

 

1.645*0.85/sqrt(n) <= 0.25

sqrt(n) >= 1.645*0.85/0.25 = 5.593

n >= 31.3  

 

So n = 32 is the minimum number of samples 

 

To get the tolerance within 0.15 ounces requires 

 

sqrt(n) >= 1.645*0.85/0.15 ~ 9.3217

n >= 86.9 

 

So n = 87 samples would be required 

 

To get the tolerance to 0.15 requires more samples because you are trying to get a better estimate of the population mean