Test and Confidence Intervals for $\mu$, ($\sigma$ unknown)

1.

According to specifications, the mean time required toinflatea rubber raft is 7.5 seconds. A random sample of 45 rafts are inflated, yeilding an average inflation time of 7.6 seconds and a standard deviation of 0.6 seconds. If our research hypothesis is that the specified time isd infact too low:

a)

Write out the null and alternative hypotheses. Define $\mu$.

b)

Calculate the value of the test statistic.

c)

Calculate the p-value. How strong is the evidence against the null hypothesis?

d)

What is your conclusion at alpha = 0.10? at alpha = 0.05? at $\alpha = 0.01$?

e)

Calculate a $95\%$ confidence interval for mean inflation time.


2.

A soft drink vending machine is set to dispence 175 ml per cup. The manufacturer is at risk, either of losing money or incuring the wrath of his customers, if the cups are are overfilled or underfilled. Ten cups of pop are dispensed with an average cup fill of 170 ml and a standard deviation of of 8.0 ml.

a)

Test whether the machine is working properly or not. State null and alternative hypothesis, value of test statistic and p-value for your test.

b)

What do you conclude at $\alpha = 0.05$? at $\alpha = 0.10$?

c)

Calculate a $95\%$ confidence interval for $\mu$, the mean cup fill.

Calculate a $90\%$ confidence interval for $\mu$.

How do these confidence intervals relate to the conclusions for the test in part b)?

d)

What assumptions must we make about the distribution of cup fill?


3.

In order to be effective, the mean life of a certain mechanical component used in a spacecraft must be greater than 1100 hours. Due to prohibitive cost of the components, only three can treated under simulated space conditions. The mean lifetime for the three components was 1173.6 hours and the standard deviation was 36.3 hours. Does the data provide sufficient evidence to conclude that the component will be effective? What level of signifigance would you choose for this test? Consider the sonsequences of a Type I error as opposed the consequences of a Type II error.


4.

A production manager noticed that the time to complete a particular job was 160 minutes. The manager made some changes to the production process in an attempt to reduce the mean time to finish the job. A sample of 2 jobs have a mean time of 148 minutes and a standard deviation of 9.7 minutes. Has the mean time to finish the job been reduced? test the relevent hypothesis.


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Jonathan Payne
1999-03-22