1. The consumption rates of two aviation fuels, A and B, are compared. The sampl
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Question
1. The consumption rates of two aviation fuels, A and B, are compared. The sample from fuel A have 10 specimens le from fuel B have 12 specimens with a mean burning rate of 21 cm/s. If the standard deviation in burning rates of the fuels are 3 cm/s and 3.5cm's for fuel A and fuel B respectively, test the hypothesis that the burning rate of fuel B is more than that of fucl A by at least 2.5cm/s. Usc = 0.05 a. Write the appropriate hypothesis b. Use P-value approach for hypothesis testing. c. Use z-test for hypothesis testing. d. Use confidence interval for hypothesis testing e. Clearly write your conclusion f. If the true difference in the mean burning rates is -:-3 cm/s, find the type ll error g. What is the minimum sample size needed to recognize the true mean burning rate of 3 cm/s for = 0.05, and = 0.05Explanation / Answer
The provided sample means are shown below:
X_1 = 18 X¯1=18 X_2 = 21X¯2=21
Also, the provided population standard deviations are:
sigma_1 = 31=3sigma_2 = 3.52=3.5
and the sample sizes are n_1 = 10n1=10 and n_2 = 12n2=12.
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho: mu_1 mu_2Ho:12
Ha: mu_1 > mu_2Ha:1>2
This corresponds to a right-tailed test, for which a z-test for two population means, with known population standard deviations will be used.
(2) Rejection Region
Based on the information provided, the significance level is lpha = 0.05=0.05, and the critical value for a right-tailed test is z_c = 1.64zc=1.64.
The rejection region for this right-tailed test is R = {z: z > 1.64}R={z:z>1.64}
(3) Test Statistics
The z-statistic is computed as follows:
z = rac{ar X_1 - ar X_2}{sqrt{sigma_1^2/n_1 + sigma_2^2/n_2}} = rac{ 18 - 21}{sqrt{ 3^2/10 + 3.5^2/12}} = -2.165z=12/n1+22/n2X¯1X¯2=32/10+3.52/121821=2.165
(4) Decision about the null hypothesis
Since it is observed that z = -2.165 le z_c = 1.64z=2.165zc=1.64, it is then concluded that the null hypothesis is not rejected.
Using the P-value approach: The p-value is p = 0.9848p=0.9848, and since p = 0.9848 ge 0.05p=0.98480.05, it is concluded that the null hypothesis is not rejected.
Confidence Interval
The 95% confidence interval for mu_1-mu_212 is -5.716 < mu_1 - mu_2 < -0.2845.716<12<0.284.
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