Suppose that a random sample of 50 cans of a particular brand of fruit juice is selected, and the amount of juice (in ounces) in each of the cans is determined. Let ? denote the mean amount of juice for the population of all cans of this brand. Suppose that this sample of 50 results in a 95% confidence interval for ? of  (7.4, 8.8). (a) Would a 90% confidence interval have been narrower or wider than the given interval? Explain your answer. A 90% confidence interval would have been narrower because the t critical value for 90% confidence is smaller than the t critical value for 95% confidence for the same sample size.A 90% confidence interval would have been narrower because the t critical value for 90% confidence is larger than the t critical value for 95% confidence for the same sample size.    A 90% confidence interval would have been wider because the t critical value for 90% confidence is smaller than the t critical value for 95% confidence for the same sample size.A 90% confidence interval would have been wider because the t critical value for 90% confidence is larger than the t critical value for 95% confidence for the same sample size. (b) Consider the following statement: There is a 95% chance that ? is between 7.4 and 8.8. Is this statement correct? Why or why not? The statement is correct. The population mean, ?, is not a constant, and therefore, we can talk about the probability that it falls within a certain interval.The statement is correct. The population mean, ?, is a constant, and therefore, we can talk about the probability that it falls within a certain interval.    The statement is not correct. The population mean, ?, is not a constant, and therefore, we cannot talk about the probability that it falls within a certain interval.The statement is not correct. The population mean, ?, is a constant, and therefore, we cannot talk about the probability that it falls within a certain interval. (c) Consider the following statement. If the process of selecting a random sample of size 50 and then calculating the corresponding 95% confidence interval is repeated 100 times, exactly 95 of the resulting intervals will include ?. Is this statement correct? Why or why not? The statement is correct. Since it is repeated more than 30 times, we can say that in 100 such samples, exactly 95 will result in confidence intervals that contain ?.The statement is correct. Since the sample size is large, we can say that in 100 such samples, exactly 95 will result in confidence intervals that contain ?.    The statement is not correct. We can only say that on average 95 out of every 100 samples will result in confidence intervals that will contain ?.The statement is not correct. We can only say that on average 5 out of every 100 samples will result in confidence intervals that will contain ?.

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 + ...