Question

A consumer research group is interested in testing an automobile manufacturer’s claim that a new economy model will travel at least 25 miles per gallon of gasoline. Provide a null and alternative hypothesis for the test. Suppose a test on 25 cars of this model indicates an average of 24 mpg, with a sample standard deviation of 3 mpg. Compute the value for the test statistic and the p-value. Please copy your R code and the result and paste them here. Suppose the significance level is 5%. Compute the critical value for the test statistic. What conclusion should we draw for the test? Provide a practical interpretation for this conclusion. Please copy your R code and the result and paste them here. Compute the critical value for the sample mean and determine the when we should reject H0 and when we should accept H0. Provide a practical interpretation of Type II error in this case. Compute the probability of committing a Type II error (denoted as β) if the actual mileage is 23 mpg as well as the power of the test. Please copy your R code and the result and paste them here.

Answer 1

The hypothesis is

H0: economy model will not travel at least 25 miles per gallon of gasoline
H1: economy model will travel at least 25 miles per gallon of gasoline

Please see the R code below

n <- 25
xbar <- 24
mu <- 25
sig <- 3

## we know that the t stat is

tstat <- (xbar - mu)/(sig/sqrt(n))

tstat

df = n-1

2*pt(-abs(tstat), df, lower=TRUE)

> tstat
[1] -1.666667
> 2*pt(-abs(tstat), df, lower=TRUE)
[1] 0.1085801

the results are p = 0.1085

as the alspha is 0.05 , hence p value > alpha and we fail to reject the null hypothesis

type2 error is when we accept the null hypothesis when we should not

Hence in this case type 2 error occurs when we conlude that economy model will not travel at least 25 miles per gallon of gasoline even though the true result was that economy model will travel at least 25 miles per gallon of gasoline