Answer:
Step-by-step explanation:
We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean
For the null hypothesis,
µ = 7463
For the alternative hypothesis,
µ ≠ 7463
This is a 2 tailed test
Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is
z = (x - µ)/(σ/√n)
Where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 7463 hours
x = 7163 hours
σ = 1080 hours
n = 81
b) z = (7163 - 7463)/(1080/√81) = - 2.5
Looking at the normal distribution table, the probability corresponding to the z score is 0.02
Recall, population mean is 7463
The difference between sample sample mean and population mean is 7463 - 7163 = 300
Since the curve is symmetrical and it is a two tailed test, the x value for the left tail is 7463 - 300 = 7163
the x value for the right tail is 7463 + 300 = 7763
These means are higher and lower than the null mean. Thus, they are evidence in favour of the alternative hypothesis. We will look at the area in both tails. Since it is showing in one tail only, we would double the area. We already got the area below z as 0.02
We would double this area to include the area in the right tail of z = 2.5. Thus
p = 0.02 × 2 = 0.04
It means that in a sample of size 81 light bulbs, we would observe a sample mean of 300 hours or more away from 7463 about 4% of the time by chance alone.
c) Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
The sample mean, x is the point estimate for the population mean.
Margin of error = z × σ/√n
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.025 = 0.975
The z score corresponding to the area on the z table is 1.96. Thus, z score for 95% confidence level is 1.96
Margin of error = 1.96 × 1080/√81
= 235.2
Confidence interval = 7163 ± 23.2
a) Since alpha, 0.05 > than the p value, 0.04, then we would reject the null hypothesis. Therefore, at a 5% level of significance, there is evidence that the mean life is different from 7463 hours
Comparing the results of a and c, it is true that the population mean life is not 7463 hours.
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