Questions 1 and 2 use the X and Y data below. You can think of this data as representing the heights of 5 individuals (labelled A,B,C,D,E) at ages 11 (X column) and 12 (Y column). Obs X Y Paired Y-X A 60 62 B 61 62 C 62 63 D 63 64 E 64 66 The means and variances for X and Y are shown below. (You can verify to practice calculation steps.) X Mean: 62, X Variance: 2.5 | Y Mean: 63.4, Y Variance: 2.8 The Y - X difference column has not been computed. You need to fill this out and compute the corresponding mean and variance (for Y-X) for question 2 1) Test whether the population mean of Y is no greater than 1/2 (0.5) unit higher than the population mean for X as the null hypothesis (Ho) vs the population mean of Y is more than 1/2 unit greater than the population mean for X as the alternative hypothesis (H1). What is the small sample t statistic for the test of Ho? Round to two decimal places (e.g., 1.23) Assume the measurements are independent (not a paired test), the variances are not equal and the degrees of freedom for the small sample test can be approximated by 5 + 5 - 2 = 8. 2) Now treat X and Y as non-independent measurements, where and the values in each row are for the same individual (at age 11 and then age 12). Compute the small sample paired-difference t statistic for the same null and alternative hypotheses. (Round your answer to 2 decimal places)

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.4: Distributions Of Data

Problem 20PFA

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Questions 1 and 2 use the X and Y data below. You can think of this data as representing the heights of 5 individuals (labelled A,B,C,D,E) at ages 11 (X column) and 12 (Y column).

Obs	X	Y
A	60	62
B	61	62
C	62	63
D	63	64
E	64	66

The means and variances for X and Y are shown below. (You can verify to practice calculation steps.)

X Mean: 62, X Variance: 2.5 | Y Mean: 63.4, Y Variance: 2.8

The Y - X difference column has not been computed. You need to fill this out and compute the corresponding mean and variance (for Y-X) for question 2

1) Test whether the population mean of Y is no greater than 1/2 (0.5) unit higher than the population mean for X as the null hypothesis (Ho) vs the population mean of Y is more than 1/2 unit greater than the population mean for X as the alternative hypothesis (H1). What is the small sample t statistic for the test of Ho? Round to two decimal places (e.g., 1.23) Assume the measurements are independent (not a paired test), the variances are not equal and the degrees of freedom for the small sample test can be approximated by 5 + 5 - 2 = 8.

2) Now treat X and Y as non-independent measurements, where and the values in each row are for the same individual (at age 11 and then age 12). Compute the small sample paired-difference t statistic for the same null and alternative hypotheses. (Round your answer to 2 decimal places)

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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