Answer:
The p-value obtained is greater than the significance level at which the test was performed, hence, we fail to reject the null hypothesis & say that there is enough evidence to conclude that the mean life of the fluorescent lights is 700 hours
Explanation:
Complete Question
fluorescent light factory that manufactures fluorescent lights that have a lifetime that is approximately normally distributed with a mean of 700 hours and a standard deviation of 45 hours. Test the hypothesis that mean is 700 hours against the alternative hypothesis that the mean is not equal to 700 hours, if a random sample of 30 bulbs has an average life of 688 hours. Use a P-value in your answer.
Solution
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.
For this question, the null hypothesis is that the mean life of the fluorescent lights is 700 hours.
The alternative hypothesis will now be that the mean life of the fluorescent lights is significantly different from 700 hours.
Mathematically,
The null hypothesis is represented as
H₀: μ = 700
The alternative hypothesis is represented as
Hₐ: μ ≠ 700
To do this test, we will use the z-distribution because no information on the population standard deviation is known
So, we compute the z-test statistic
z = (x - μ₀)/σₓ
x = sample mean = 688 hours
μ₀ = 700 hours
σₓ = standard error = (σ/√n)
σ = standard deviation = 45 hours
n = Sample size = 30
σₓ = (45/√30) = 8.2158383626 = 8.216
z = (688 - 700) ÷ 8.216
z = 1.46
checking the tables for the p-value of this z-statistic
Significance level = 0.05 (This is used when significance level isn't given)
The hypothesis test uses a two-tailed condition because we're testing in two directions.
p-value (for z = 1.46, at 0.05 significance level, with a two tailed condition) = 0.14429
The interpretation of p-values is that
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
So, for this question, significance level = 0.05
p-value = 0.14429
0.14429 > 0.05
Hence,
p-value > significance level
This means that we fail to reject the null hypothesis & say that there is enough evidence to conclude that the mean life of the fluorescent lights is 700 hours.
Hope this Helps!!