What Is the Probability of a Type 2 Error in Fuel Efficiency Claims?

[Pinto09]

1) A car producer claims a certain type of car is fuel-saving and can be driven 25 miles/gallon and a st dev of 2.8 miles/gallon.
An analyst wants to test this claim and wants a 1% risk of concluding the statement is false when it in fact is true. He chooses 30 observations.

Calculate the probability for type 2 error if the true mean is 24?

I tried:
X-bar critical = 26.32 and 23.68 and got P(-0.62 < Z < 4.53) = F(0.62)

But the book's answer is 0.64?


2)
A company considers launching a new product if the proportion of customers who will buy it is at least 0.2.
A sample of 100 was taken. The null hypothesis (P =0.2) is rejected if the smaple proprtion is at least o.27.

The following requirements must be met:
P(not rejecting H0) = at least 99% if true proportion is 0.17
P(rejecting H0) = at least 99% if true proportion is 0.27

e) What sample size is required (at least)?
Book: 365
f) What is the significance level?
Book: 17%

How do you solve this??

 

[mighty2000]

1) To calculate the probability of type 2 error, we first need to determine the critical z-scores for a 1% risk of concluding the statement is false when it is actually true. This can be determined using a z-table or a statistical software. In this case, the critical z-scores are -2.326 and 2.326.

Next, we need to calculate the standard error of the mean using the given standard deviation and sample size. The formula for standard error of the mean is given by SE = σ/√n, where σ is the standard deviation and n is the sample size. In this case, SE = 2.8/√30 = 0.511.

Now, we can calculate the z-score for the true mean of 24 using the formula z = (x̄ - μ)/SE, where x̄ is the sample mean and μ is the true mean. In this case, z = (24-25)/0.511 = -1.96.

Finally, we can calculate the probability of type 2 error using the z-score calculated in the previous step and the critical z-scores. The probability of type 2 error is given by P(Z < -2.326) + P(Z > 2.326) = 0.0078 + 0.0078 = 0.0156, which is approximately 1.56%.

2) To determine the sample size required, we can use the formula n = (Zα/2)^2 * p̂ * (1-p̂)/d^2, where Zα/2 is the critical z-score for the desired confidence level, p̂ is the expected proportion, and d is the margin of error. In this case, Zα/2 = 2.576 for a 99% confidence level, p̂ = 0.27, and d = 0.07 (difference between 0.27 and 0.2).

Substituting these values in the formula, we get n = (2.576)^2 * 0.27 * (1-0.27)/(0.07)^2 = 365. Therefore, a sample size of at least 365 is required to meet the given requirements.

To determine the significance level, we need to find the critical z-score for a 99% confidence level. This can be done using