onsider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ1 and σ2 are unknown but we have reason to believe that σ1 = σ2, we can pool the standard deviations. Using sample sizes n1 and n2, the sample test statistic x1 − x2 has a Student's t distribution where t = x1 − x2 s 1 n1 + 1 n2 with degrees of freedom d.f. = n1 + n2 − 2 and where the pooled standard deviation s is s = (n1 − 1)s12 + (n2 − 1)s22 n1 + n2 − 2 Note: With statistical software, select the pooled variance or equal variance options. (a) There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal. Consider a report regarding average incidence of fox rabies in two regions. For region I, n1 = 14,x1 ≈ 4.75, and s1 = 2.84 and for region II, n2 = 13, x2 ≈ 3.95, and s2 = 2.41. The two sample standard deviations are sufficiently close that we can assume σ1 = σ2. Use the method of pooled standard deviation to consider the report, testing if there is a difference in population mean average incidence of rabies at the 5% level of significance. (Round your answer to three decimal places.) t =

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Description: onsider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ1 and σ2 are unknown but we have reason to believe that σ1 = σ2, we can pool the standa

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 6CR

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onsider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ₁ and σ₂ are unknown but we have reason to believe that σ₁ = σ₂, we can pool the standard deviations. Using sample sizes n₁ and n₂, the sample test statistic x₁ − x₂ has a Student's t distribution where

t =

x₁ − x₂

n₁

n₂

with degrees of freedom d.f. = n₁ + n₂ − 2

and where the pooled standard deviation s is

s =

(n₁ − 1)s₁² + (n₂ − 1)s₂²

n₁ + n₂ − 2

Note: With statistical software, select the pooled variance or equal variance options.

(a) There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal. Consider a report regarding average incidence of fox rabies in two regions. For region I, n₁ = 14,x₁ ≈ 4.75, and s₁ = 2.84 and for region II, n₂ = 13, x₂ ≈ 3.95, and s₂ = 2.41. The two sample standard deviations are sufficiently close that we can assume σ₁ = σ₂. Use the method of pooled standard deviation to consider the report, testing if there is a difference in population mean average incidence of rabies at the 5% level of significance. (Round your answer to three decimal places.)

t =

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 + ...

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