Question

The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, with σ = 7. A random sample of 92 matchboxes shows the average number of matches per box to be 42.7. Using a 1% level of significance, can you say that the average number of matches per box is more than 40?

What are we testing in this problem?

single mean

single proportion

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 40; H₁: p > 40H₀:

p = 40; H₁: p < 40    H₀: μ = 40; H₁: μ < 40H₀:

p = 40; H₁: p ≠ 40H₀:

μ = 40; H₁: μ ≠ 40H₀

: μ = 40; H₁: μ > 40

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal, since we assume that x has a normal distribution with unknown σ.

The Student's t, since we assume that x has a normal distribution with unknown σ.    

The Student's t, since we assume that x has a normal distribution with known σ.

The standard normal, since we assume that x has a normal distribution with known σ.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the average number of matches per box is now greater than 40.There is insufficient evidence at the 0.01 level to conclude that the average number of matches per box is now greater than 40.    

Answer 1

a) We are testing the sample mean in this problem.

Level of significance= 0.01

We are testing,

H0: u= 40 vs H1: u> 40

b) Since population SD is known, we will use standard normal distribution. So option d is correct.

Test statistic: (sample mean-40)/(population SD/√n) = (42.7-40)/(7/√92) = 0.96 (rounded to two decimals)

c) p-value of this one sided z test:

P(z>0.96) = 0.1685 (from the standard normal distribution tables)

So 0.125<p-value <0.250 is correct

We will have a normal curve here

d) Since the p-value of our test>significance level of 0.01, we have sufficient evidence to Reject H0. So option a) is correct here.

e) there is sufficient evidence to conclude that at the 1% significance level, u> 40

So option a is correct.