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# Hypothesis Testing

1.You are visiting Dad's Root Beer Plant in Oakland. You are asked by the plant supervisor to determine if the Dad's Brand soda machines are correctly dispensing 32 ounces of root beer. In the past, the equipment was assumed to be not working properly if the cans were over-filled or under-filled. You observe the machine filling 9 bottles and collected the data. You are 95% sure that the machine is filling the bottles correctly. The amount dispersed is assumed to be normally distributed:

32.12 32.04 32.04
32.13 32.05 32.05
32.22 31.89 32.20

2. You are given the following information, find the Z-statistic: (95% level of confidence) based on the Z-test

Xbar = 980
U = 1000
Sample stdev = 50
N = 36

3. The average running time for current Broadway shows is 2 hours and 12 minutes. A producer in another city claims that the length of time of productions in his city is the same. He samples 8 shows and finds the time to be 2 hours and 5 minutes with a standard deviation of 11 minutes. At 95% level of confidence is the producer correct (NY Times Arts and Leisure)? Keep in mind, your numbers must be in minutes.

4.Find the T-critical if N = 22 at 95% level of confidence and 90% level of confidence

5.The average salary of graduates entering the actuarial field is reported to be \$40,000. You sample 20 graduates and find the average salary to be \$43,228 with a standard deviation of \$4,000. At a 95% level of confidence is the statement regarding the average salary of \$40K an accurate statement?

#### Solution Preview

Please refer attached file for complete solutions. Work done with the help of equation writer is missing here.

Solutions:

1.
Ho: Average dispensing quantity is 32 ounces. (i.e. )

H1: Average dispensing quantity is not equal to 32 ounces. (i.e. )
It's a two tail test.
Level of significance =95%

Here sample size is less than 30 and population standard deviation is not given (We are given that population is approximately normal) we will use t-statistics.

Sample size=n=9
Degrees of freedom=9-1=8
Sample mean=Xbar=(32.12+32.04+32.04+32.13+32.05+32.05+32.22+31.89+32.20)/9=32.0822

Sample std deviation=s=sqrt{[(32.12-32.0822)^2+(32.04-32.0822)^2+(32.04-32.0822)^2+(32.13-32.0822)^2+?(32.05-32.0822)^2+(32.05-32.0822)^2+(32.22-32.0822)^2+(31.89-32.0822)^2+(32.20-32.0822)^2]/(9-1)}= 0.0995

Since population standard deviation is not given, we will estimate it with sample standard deviation.

Estimated standard error =estimated population standard deviation/sqrt(sample size)
=0.0995/sqrt(9)= 0.03316

To find out t-critical, refer t-distribution tables and look for combined area=(1-0.95%=0.05) with 8 degrees of freedom
t-critical= 2.306
(observed value of t should lie ...

#### Solution Summary

There are five problems. Solutions to these problems explain the steps to find test statistics and all the steps needed in testing the associated claims.

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