Question

A possible measure of efficiency of an engine is the power it produces for each unit of volume. The efficiency may be measured for each car using the ratio between the power and the volume of the engine. Consider, for each car, the ratio between the power of the engine measured in horsepower and the volume of the engine measured by the volume swept by all the pistons inside the cylinders in cubic inches. Call this variable Y. In Chapter 12 we will present a statistical test for testing that the expectation of a variable is equal to a given value, say 1. The test statistic for this problem is given by: T = [Y-1]/[S/ √n], where Y is the sample average, S2 is the sample variance and n is the sample size. In the following questions you are required to determine the region that contains 95% of the values of the test statistic if indeed the ratio between power and size of engine is equal to 1 and to check whether or not the evaluation of the test statistic for the "cars" data set falls within this

region.

Let Y₁, Y₂, ..., Y₂₀₃ be a sample of Normal random variables with expectation μ=1 and variance σ²=0.04. (Namely, standard deviation of σ=0.2.) The region that contains 95% of the distribution of the statistic T = [Y-1]/[S/ √203] is best described by:

Select one:

a. [-19.7,19.7]

b. [88.3,127.7]

c. [-1.97,1.97]

d. [106,110]

Consider a variable by the name "y" that contains the ratio between the power of the engine and the size of the engine in the data set "cars". Use the fact that the sample average of the variable is y=0.8237 and and the sample standard deviation is s= 0.1833 in order to compute the value of the test statistic for this variable. The value of the statistic for the observed data set "cars" is:

Answer 1

Result:

A possible measure of efficiency of an engine is the power it produces for each unit of volume. The efficiency may be measured for each car using the ratio between the power and the volume of the engine. Consider, for each car, the ratio between the power of the engine measured in horsepower and the volume of the engine measured by the volume swept by all the pistons inside the cylinders in cubic inches. Call this variable Y. In Chapter 12 we will present a statistical test for testing that the expectation of a variable is equal to a given value, say 1. The test statistic for this problem is given by: T = [Y-1]/[S/ √n], where Y is the sample average, S2 is the sample variance and n is the sample size. In the following questions you are required to determine the region that contains 95% of the values of the test statistic if indeed the ratio between power and size of engine is equal to 1 and to check whether or not the evaluation of the test statistic for the "cars" data set falls within this region.

Let Y₁, Y₂, ..., Y₂₀₃ be a sample of Normal random variables with expectation μ=1 and variance σ²=0.04. (Namely, standard deviation of σ=0.2.) The region that contains 95% of the distribution of the statistic T = [Y-1]/[S/ √203] is best described by:

Select one:

a. [-19.7,19.7]

b. [88.3,127.7]

Correct option: c. [-1.97,1.97]

d. [106,110]

( critical t value of t with 202 df at 95% level is 1.97)

Consider a variable by the name "y" that contains the ratio between the power of the engine and the size of the engine in the data set "cars". Use the fact that the sample average of the variable is y=0.8237 and the sample standard deviation is s= 0.1833 in order to compute the value of the test statistic for this variable. The value of the statistic for the observed data set "cars" is:

Test statistic

= -13.7037