Question

A machine that is programmed to package 3.30 pounds of cereal is being tested for its accuracy. In a sample of 49 cereal boxes, the sample mean filling weight is calculated as 3.39 pounds. The population standard deviation is known to be 0.14 pound. [You may find it useful to reference the z table.]

a-1. Identify the relevant parameter of interest for these quantitative data.

-The parameter of interest is the proportion filling weight of all cereal packages.

-The parameter of interest is the average filling weight of all cereal packages.

a-2. Compute its point estimate as well as the margin of error with 99% confidence. (Round intermediate calculations to at least 4 decimal places. Round "z" value to 3 decimal places and final answers to 2 decimal places.)

b-1. Calculate the 99% confidence interval. (Use rounded margin of error. Round your final answers to 2 decimal places.)

b-2. Can we conclude that the packaging machine is operating improperly?

Yes, since the confidence interval contains the target filling weight of 3.30.

No, since the confidence interval does not contain the target filling weight of 3.30.

No, since the confidence interval contains the target filling weight of 3.30.

Yes, since the confidence interval does not contain the target filling weight of 3.30.

c. How large a sample must we take if we want the margin of error to be at most 0.03 pound with 99% confidence? (Round intermediate calculations to at least 4 decimal places. Round "z" value to 3 decimal places and round up your final answer to the next whole number.)

Answer 1

Solution:

Given:

Sample size = n = 49

Sample mean =   pounds

Population standard deviation = pounds

Part a-1. Identify the relevant parameter of interest for these quantitative data.

Since we have to test for mean filling weight , the relevant parameter of interest is:

-The parameter of interest is the average filling weight of all cereal packages.

Part a-2. Compute its point estimate as well as the margin of error with 99% confidence.

Point estimate =  Sample mean =   pounds

The margin of error is:

Z_c is z critical value for c = 0.99 confidence level.

Find Area = ( 1+c)/2 = ( 1 + 0.99 ) / 2 = 1.99 /2 = 0.9950

Thus look in z table for Area = 0.9950 or its closest area and find corresponding z critical value.

From above table we can see area 0.9950 is in between 0.9949 and 0.9951 and both are at same distance from 0.9950, Hence corresponding z values are 2.57 and 2.58

Thus average of both z values is 2.575

Thus Z_c = 2.575

Thus

Part b-1. Calculate the 99% confidence interval.

Part b-2. Can we conclude that the packaging machine is operating improperly?

Yes, since the confidence interval does not contain the target filling weight of 3.30.

Part c. How large a sample must we take if we want the margin of error to be at most 0.03 pound with 99% confidence?

E = 0.03