The​ quality-control manager at a compact fluorescent light bulb​ (CFL) factory needs to determine whether the population mean life of a large shipment of CFLs is equal to​ 7,497 hours. The population standard deviation is​ 1,080 hours. A random sample of 81 light bulbs indicates a sample mean life of​ 7,197 hours.   a. Let μ be the population mean. Determine the null​ hypothesis, H0​, and the alternative​ hypothesis, H1.                   A. H0 : μ = 7,197 and H1 : μ ≠ 7,197, because the sample mean is always used in hypothesis testing                   B. H0 : μ = 7,497 and H1 : X = 7,197, because the population mean is used in H0 and the sample                         mean           in H1                  C. H0 : μ ≠ 7,497 and H1 : μ = 7,497​, because H0 never uses the equal symbol                  D. H0 : μ = 7,497 and H1 : μ ≠ 7,497 because the​ "goal" or​ "historical data" is always used when stating              the hypotheses   What is the value of the test statistic? ​(Round to two decimal places to the right of the decimal point as​ needed.)     See fig 1.   What​ is/are the critical​ value(s)? ​(Round to two decimal places to the right of the decimal point as needed. Use a comma to separate answers as​ needed.) A. ​-1.96 and​ +1.96 found using±​NORM.S.INV(0.05/2) B.​-1.96 found using​ (NORM.S.INV(0.05/2) C.​+1.96 found using​ -(NORM.S.INV(0.05/2)) D.​-1.64 and​ +1.64 found using±​NORM.S.INV(0.05)   What is the final​ conclusion?   A. Fail to reject H0. There is sufficient evidence to prove that the mean life is different from​ 7,497 hours. B. Fail to reject H0. There is not sufficient evidence to prove that the mean life is different from​ 7,497 hours. C. Reject H0. There is sufficient evidence to prove that the mean life is different from​ 7,497 hours. D. Reject H0. There is not sufficient evidence to prove that the mean life is different from​ 7,497 hours.   b. What is the​ p-value? ​(Round to three decimal places to the right of the decimal point as​ needed.) Interpret the meaning of the​ p-value. Choose the correct answer below.   A. Fail to reject H0. There is sufficient evidence to prove that the mean life is different from​ 7,497 hours. B. Reject H0. There is sufficient evidence to prove that the mean life is different from​ 7,497 hours. C. Reject H0. There is not sufficient evidence to prove that the mean life is different from​ 7,497 hours. D. Fail to reject H0. There is not sufficient evidence to prove that the mean life is different from​ 7,497 hours.   c. Construct a​ 95% confidence interval estimate of the population mean life of the light bulbs.  ​(Round to one decimal place as​ needed.) A. 6,961.8 ≤ μ ≤ 7,432.2​, found using​ 7,197± ​CONFIDENCE.NORM(0.05,1080,81) B. 7,261.8 ≤ μ ≤ 7,732.2​, found using​ 7,497±​CONFIDENCE.NORM(0.05,1080,81) C. 6,928.0 ≤ μ ≤ 7,466.0​, found using​ 7,197±​CONFIDENCE.NORM(0.05/2,1080,81) D. 7,170.9 ≤ μ ≤ 7,223.1​, found using​ 7,197±CONFIDENCE.NORM(0.05,120,81)   d. Compare the results of​ (a) and​ (c). What conclusions do you​ reach? A.The results of​ (a) and​ (c) are not the same: there is not sufficient evidence to prove that the mean life is different from​ 7,497 hours. B.The results of​ (a) and​ (c) are not the​ same: there is sufficient evidence to prove that the mean life is different from​ 7,497 hours. C.The results of​ (a) and​ (c) are the​ same: there is not sufficient evidence to prove that the mean life is different from​ 7,497 hours. D.The results of​ (a) and​ (c) are the​ same: there is sufficient evidence to prove that the mean life is different from​ 7,497 hours

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...