Perform a one-sample  z -test for a population mean using the  P -value approach. Be sure to state the hypotheses and the significance level, to compute the value of the test statistic, to obtain the  P -value, and to state your conclusion. In the past, the mean running time for a certain type of flashlight battery has been  8.2  hours. The manufacturer has introduced a change in the production method which he hopes has increased the mean running time. The mean running time for a random sample of  40  light bulbs was  8.4  hours. Do the data provide sufficient evidence to conclude that the mean running time of all light bulbs,  μ , has increased from the previous mean of  8.2  hours? Perform the appropriate hypothesis test using a significance level of  0.05 . Assume that  σ  =  0.5  hours.     H0  :  =μ8.2  hours H1  :  >μ8.2  hours =α0.05 =z2.62 P -value  =0.0044 Reject  H0  . At the  5%  significance level, the data provide sufficient evidence to conclude that the mean running time of all light bulbs,  μ , has increased from the previous mean of  8.2  hours.     H0  :  =μ8.4  hours H1  :  >μ8.4  hours =α0.05 =z2.62 P -value  =0.0044 Reject  H0  . At the  5%  significance level, the data provide sufficient evidence to conclude that the mean running time of all light bulbs,  μ , has increased from the previous mean of  8.4  hours.     H0  :  =μ8.2  hours H1  :  >μ8.2  hours =α0.05 =z2.53 P -value  =0.0057 Reject  H0  . At the  5%  significance level, the data provide sufficient evidence to conclude that the mean running time of all light bulbs,  μ , has increased from the previous mean of  8.2  hours.     H0  :  =μ8.4  hours H1  :  >μ8.4  hours =α0.05 =z2.53 P -value  =0.0057 Reject  H0  . At the  5%  significance level, the data provide sufficient evidence to conclude that the mean running time of all light bulbs,  μ , has increased from the previous mean of  8.4  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 + ...