A marine biologist claims that the mean length of mature female pink seaperch is different in fall and winter. A sample of 12 mature female pink seaperch collected in fall has a mean length of 115 millimeters and a standard deviation of 11 millimeters. A sample of 10 mature female pink seaperch collected in winter has a mean length of 106 millimeters and a standard deviation of 8 millimeters. At a = 0.01, can you support the marine biologist's claim? Assume the population variances are equal. Assume the samples are random and independent, and the populations are normally distributed. Complete parts (a) through (e) below. (a) Identify the claim and state Ho and Ha. Which is the correct claim below? O A. "The mean length of mature female pink seaperch is greater in the fall than in the winter." O B. "The mean length of mature female pink seaperch is greater in the winter than in the fall." OC. "The mean length of mature female pink seaperch is the same in fall and winter." O D. "The mean length of mature female pink seaperch is different in fall and winter." What are Ho and H,? The null hypothesis, Ho, is V The alternative hypothesis, Ha, is Which hypothesis is the claim? The alternative hypothesis, Ha O The null hypothesis, Ho (b) Find the critical value(s) and identify the rejection region(s). Enter the critical value(s) below. (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.) Select the correct rejection region(s) below. O A. t< -to O B. t< -to, t>l OC. -toto (c) Find the standardized test statistic. t%=(Type an integer or decimal rounded to three decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. v the null hypothesis. (e) Interpret the decision in the context of the original claim. At the 1% significance level, enough evidence to support the claim that the mean lengths of mature female pink seaperch are different in fall and winter.

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 + ...