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 92 matchboxes shows the average number of matches per box to be 42.9. 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 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.    

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.

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.

(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

Solution:
Given in the question
the population mean ()= 40
Population standard deviation() = 8
Sample size n =92
Sample mean = 42.9
Level of significance = 0.01
In this question, we are testing Single mean
Solution(a)
Level of significance = 0.01
The claim is that the average number of matches per box is more than 40
So Null and alternate hypothesis can be written as
Null hypothesis H0: = 40
Alternate hypothesis Ha: >40
Solution(b)
Here we will use standard normal since we will use assume that X has a normal distribution with known .
as the sample size is large enough and the population standard deviation is known.
Sample test statistic value can be calculated as
Test statistic = (Xbar - )//sqrt(n) = (42.9 - 40)/8/sqrt(92) = 3.48
Solution(c)
P-value from Z table and this is right tailed test is 0.000251
So p-value is less than .005 i.e P-value<0.005
Solution(d)
Here we can see that we can reject null hypothesis H0, as the p-value is less than alpha value. At alpha = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
Solution(e)
The final conclusion is that there is sufficient evidence at the 0.01 level to conclude that the average number of matches per box is greater than 40.