Question

The company has two machines which are used to fill bottles with the carbonated beverage. The actual amount filled in each bottle is assumed to have a normal distribution. For Machine 1, the standard deviation is known to be s₁=0.015 and for Machine 2 it is known to be s₂=0.018. The quality control department wants to check if the two machines fill the same net volume on average. They randomly collect 10 bottles which were filled by Machine 1, and ten which were filled by Machine 2, and measure the actual volume. The measurements are given here:

Machine 1: 16.03 16.04 16.05 16.05 16.02 16.01 15.96 15.98 16.02 15.99

Machine 2: 16.02 15.97 15.96 16.01 15.99 16.03 16.04 16.02 16.01 16

Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.

Show the formula for the test statistic and compute its value.

What is distribution of the test statistic under the null hypothesis?

Using a=0.05, what is your conclusion?

Compute the p-value.

Compute a 95% confidence interval for the difference between the mean fill volume for the two machines. Interpret your result.

Answer 1

Answer:

Based on the given data:

• Actual amount filled in bottle has a normal distribution
• Machine 1, Standard deviation = 0.015
• Machine 2, Standard deviation = 0.018
• Sample Size (bottles filled by machine 1) n₁ = 10
• Sample Size (bottles filled by machine 2) n₂ = 10

	Machine 1	Machine 2
	16.03	16.02
	16.04	15.97
	16.05	15.96
	16.05	16.01
	16.02	15.99
	16.01	16.03
	15.96	16.04
	15.98	16.02
	16.02	16.01
	15.99	16
Mean	16.02	16.01
Std.dev	0.030	0.025

Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.

Claim to test: if the two machines fill the same net volume on average.
Testing Hypothesis about difference between two population means

• Null Hypothesis,
• Alternate Hypothesis,

Show the formula for the test statistic and compute its value.

• In this case, since the standard deviations for population is known, therefore Z-test will be used.
• 

What is distribution of the test statistic under the null hypothesis?

• Since population standard deviation is known, therefore standardized Z-test is used. The test statistic is assumed to have a normal distribution.

Using a=0.05, what is your conclusion?

• P-value corresponding to Z-value of 1.3496 for a two-tailed test is 0.1771. Since P-value (0.1771) is more than the significance level , therefore Null Hypothesis is not rejected. There is not sufficient evidence to claim that the two population means are different.

Compute the p-value.

• P-value corresponding to Z-value of 1.3496 for a two-tailed test is 0.1771.

Compute a 95% confidence interval for the difference between the mean fill volume for the two machines. Interpret your result.

• 95% confidence interval for the difference between the mean fill volume for the two machines is:
•