Question

The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, with σ = 8. A random sample of 90 matchboxes shows the average number of matches per box to be 43.0. 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 proportion single mean (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 40; H1: μ ≠ 40 H0: p = 40; H1: p > 40 H0: μ = 40; H1: μ > 40 H0: μ = 40; H1: μ < 40 H0: p = 40; H1: p < 40 H0: p = 40; H1: p ≠ 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 known 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 unknown 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 > 0.250 0.125 < P-value < 0.250 0.050 < P-value < 0.125 0.025 < P-value < 0.050 0.005 < P-value < 0.025 P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (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)

level of significance = 0.01

The null and alternate hypotheses are,

H0: μ = 40; H1: μ > 40 H0: μ = 40;

(b)

As, we know the population standard deviation and the sample size us greater than 30, we assume the sampling distribution of the mean to be Normal distribution.

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

Standard error of mean = = 8 / = 0.843274

Test statistic, z = ( - ) / Std Error = (43 - 40) / 0.843274 = 3.56

(c)

P-value = P(z > 3.56) = 0.0002

P-value < 0.005

The sampling distribution will be normal curve with small shaded area at the extreme right of the curve corresponding to the P-value.

(d)

Since p-value < 0.01 significance level, we reject the null hypothesis and conclude that there is significant evidence that the average number of matches per box is more than 40.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e)

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