Question

An energy conservation group claims that the mean residential electricity consumption in Boston is greater than 1300 kwh/household. To test this claim, a random sample of 200 households is selected. The resulting sample mean is 1325 kwh and the sample standard deviation is 185 kwh. (4 pts each)

a. State your hypotheses and explain your reasons for this formulation.

b. Is there evidence to support the claim at α = 0.05?

c. Find the P-value of the test in Part (b).

d. What is β, the Type II Error, if the true population mean is 1320?

Answer 1

for testing the population mean(H) (a) Thez -test We have to perform right-tail hypothesis test. Since, the populati on standard deviation is known we must use z- test Step1: The sample size isn 200 The sample mean is X= 1325 kwh The population standard deviation is σ-185 kwh The level of signific (b) The null and alternative hypotheses are Ho : μ-1300 (The population mean is equal to μ-1300) M : μ > 1300 (The population mean is greater than μ-1300) Step 2: The sample size is n = 200, we decide to use α = 0.05 Step3: The table value of the z- test statistic is Zoos-1.645 Reject Ho if Z >1.645 otherwise do not reject Ho Step4 The summary statistics. n 200, X 1325 kwh, ơ-185 kwh The formula for calculating the z - test stati stic is 1325-1300 185 /200 25 185 14.14214 25 13.08148 1911099 1.911099 If the test statistic is z,-191 10j, zou" 1 645 we reject the null hypothesis Ho (c) Step5 Finding the P- value using the test stati stic is calculated in the part (b) and it is given P-value-P(z-z) (1-P(z <1.911099)) Since, on using Excel2010 function 1-0.972004) normsdist ( z = (0.027996) P-value 0.028| < α-0.05. We reject the null hypothesis Ho Step6: Since, the test statistic is z 19111>2oo5 1645. We reject the null hypothesis Ho There is sufficient evi dence to reject the null hypothesis Ho. Hence, we can conclude that the populati on mean resi denti al electricity consumption in Boston is greater than 1300 kwh/household.

d) The level of significance is α=0.05 The true population mean is μ 1320 The population mean is μ0 1300 The mean difference is δ-μ_Ao → 1320-1300 20 δ= 20 The sample size is n= 200 The standard deviation is ơ 185 The formula for calculating the Type I error is 20 200 20,200 (ha,- )-Ф(7m- ) 20(14.14214) 20 (14.14214) 0.0512 185 185 282.8427-1.959964 282.8427 185 → Φ(1.959964-1.52888). Φ(-1.959964-1.52888) Since, on using Excel2010 function: → Φ(0.431 084)-D(-3.48884) |=normasdist(z) 0.17361-0.000243 0.17386 β$$ 1-0, 1739 Therefore, the Type II error, if the true populati on mean is 1320 is approximately 0,1739