Module 3 Quiz

Question 1

The life in hours of a battery is known to be approximately normally distributed with standard deviation σ = 1.25 hours. A random sample of 10 batteries has a mean life of x¯ = 40.5 hours.

A. Is there evidence to support the claim that battery life exceeds 40 hours? Use α=0.05.
B. What is the P-value for the test in part A?
C. What is the β-error for the text in part B if the true mean life is 42 hours?
D. What sample size would be required to ensure that β does not exceed 0.10 if the true mean is 44 hours?
E. Explain how you could answer the question in part A by calculating an appropriate confidence bound on battery life.

Given

σ = 1.25 hours
n = 10 batteries
x¯ = 40.5 hours

A.

figure 1.

As seen above, the P-value is greater than α, therefore we fail to reject the null hypothesis. There is no evidence to support the claim that the battery life exceeds 40 hours.

B.

From figure 1, it can be seen that the P-value attained for Part A is 0.10383, or simply 0.10.

C.

figure 2.

As seen in figure 2, the β-error if the true mean life is 42 hours, is 0.0032.

D.

figure 3.

In figure 3, it can be seen by solving the equation, we get an approximation of 1 battery so that β does not exceed 0.10 if the mean is 44 hours.

E.

According to Rebecca Bevans, the confidence interval is “the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again.”

figure 4.

Confidence Interval: (39.8498 < 40 < 41.1502)

From the solution in figure 4, it can be seen that 40 falls within the confidence interval, therefore we have insufficient evidence to reject the null hypothesis.

Source: https://www.scribbr.com/statistics/confidence-interval/

Question 2

Brand A gasoline was used in 16 similar automobiles under identical conditions. The corresponding sample of 16 values (miles per gallon) had mean 19.6 and standard deviation 0.4. Under the same conditions, high-power brand B gasoline gave a sample of 16 values with mean 20.2 and standard deviation 0.6. Is the mileage of B significantly better than that of A? Assume normality. Test the hypothesis using both P-value and fixed significance level with α=0.05 approaches (if possible).

Given

A: n = 16, x¯ = 19.6, σ = 0.4
B: n = 16, x¯ = 20.2, σ = 0.6
α = 0.05

Seven-Step Hypothesis-Testing Procedure

1. Parameter of interest: The parameter of interest is the mileage mean of the two brands of gasolines, A and B, μ1 and μ2 respectively.

2. Null hypothesis: H0: μ1 = μ2

3. Alternative Hypothesis: H1: μ1 < μ2

4. Test Statistic:

figure 1.

5. Reject H0 if: P-value is less than α = 0.05.

6. Computations:

figure 2.

7. Conclusions: Since the P-value is 0.00043 which is less than α = 0.05, we reject the null hypothesis. Therefore, it is evident that the mileage of brand B is significantly better than that of brand A.