Question

you are an analyst working for the new Joint High Speed Vessel (JHSV) program office. Several tests have been conducted on two different types of experimental test vessels (denoted simply “1” and “2”) to determine their performance characteristics under various loading and sea state conditions.

With detailed data on the fuel consumption of vessel "1," and with the Navy's new focus on energy efficiency, the program manager wants to test whether the vessel beats the design specs in terms of mean hourly fuel consumption. You decide to conduct a large sample hypothesis test of the data with the goal of conclusively demonstrating, if possible, that the data support the claim that the mean hourly fuel consumption is less than 50 gph (gallons per hour) at 35 knots. Given that for 36 (independent) hours of operation at 35 kts you observe y-bar 49.5 gph with s=2 gph, and using a significance level of a=0.05:

a. Write out the null and alternative hypotheses.

b. Calculate the test statistic.

c. Calculate and state the rejection region or p-value.

d. Conduct the test and state the outcome. State the outcome both in terms of accepting or rejecting the null hypothesis and then in terms of what the result means in the context of this particular problem.

Answer 1

Let gph be the true mean hourly fuel consumption at 35 knots. We want to test the clim that the mean hourly fuel consumption is less than 50 gph (gallons per hour) at 35 knots. That is we want to test the claim that .

a. Write out the null and alternative hypotheses.

The hypotheses are

b. Calculate the test statistic.

We have the following information from the sample

n=36 is the sample size

gph is the sample mean fuel consumption per hour

gph is the sample standard deviation of fuel consumption per hour

We will estimate the population standard deviation using the sample

The estimated standard error of mean fuel consumption per hour is

The hypothesized value of mean fuel consumption is

The sample size is greater than 30. Hence we can use large sample analysis. That is using the central limit theorem, we can say that the sampling distribution of mean is normal.

The test statistics is

ans: the test statistic is -1.5

c. Calculate and state the rejection region or p-value.

This is a left tail test (The alternative hypothesis is "less than")

The left tail critical value is obtained using

Using the standard normal tables we get that for z=1.645, P(Z<1.645) = 0.95 or P(Z<-1.645) = 0.05

The critical value is -1.645. We will reject the null hypothesis if the test statistics is less than -1.645

ans: The rejection region is <-1.645

This is a left tail test. The p-value is

The p-value is 0.0668. We will reject the null hypothesis if the p-value is less than the significance level .

ans: p-value is 0.0668

d. Conduct the test and state the outcome. State the outcome both in terms of accepting or rejecting the null hypothesis and then in terms of what the result means in the context of this particular problem.

Using the critical value: We will reject the null hypothesis if the test statistics is less than -1.645. Here the test statistics is -1.5 and it is not less than -1.645. Hence we do not reject the null hypothesis.

Using the p-value. We will reject the null hypothesis if the p-value is less than the significance level 0.05. Here, the p-value is 0.0668 and it is not less than the significance level, 0.05. Hence we do not reject the null hypothesis.

ans: We fail to reject the null hypothesis

ans: We conclude that there is no sufficient evidence to support the claim that the mean hourly fuel consumption is less than 50 gph (gallons per hour) at 35 knots. The the vessel does not beat the design specs in terms of mean hourly fuel consumption.