Question

The following are a few problems on hypothesis testing. The corresponding confidence interval
(CI) question is also asked. This is a good practice for students to see how hypothesis testing
results reflect in the confidence interval estimates. In the first problem, the steps are given to get
the hypothesis testing results and the CI estimates. It is a good practice to follow the steps. Try to
use the same steps in the subsequent problems.

1. The life span of 100 W light bulbs manufactured by a particular company follows a normal
distribution with a standard deviation of 120 hours and its true average half-life is guaranteed
under warranty for a minimum of 800 hours. At random, a sample of 50 bulbs from a lot is
selected and it is revealed that the average half-life is 750 hours. With a significance level of
0.01, should the lot be rejected by not honoring the warranty? Give the 99% upper confidence
bound for the true average half-life. Does the confidence limit reflect the same result as the
hypothesis test?

Identify the parameter under study.
Steps in Hypothesis testing:

STEP 1 Set up the hypothesis
STEP 2: Get the sample information and find the test statistic value
STEP 3: Find the Rejection Region.
STEP 4: Compare and conclude.
STEP 5: Interpret
For the 95% confidence interval:

STEP 1: Find the point estimate/statistic:
STEP 2: Find the SE (x)
STEP 3: Find the multiplier
STEP 4: Construct the CI
STEP 5: Interpret

Reflection on the hypothesis test results and CI estimate:
2. A manufacturer of electric lamps is testing a new production method that will be considered
acceptable if the lamps produced by this method result in a normal population with an average
life of 2,400 hours and a standard deviation equal to 300. A sample of 100 lamps produced by
this method has an average life of 2,320 hours. Can the hypothesis of validity for the new
manufacturing process be accepted with a probability of type I error being 5%? Construct the
95% confidence interval for the average life of electric lamps and relate it to the result of the
hypothesis test.

3. The quality control division of a factory that manufactures batteries suspects defects in the
production of a model of mobile phone battery which results in a lower life for the product.
Until now, the time duration in phone conversation for the battery followed a normal
distribution with a mean of 300 minutes. However, in an inspection of the last batch produced
before sending it to market, it was found that the average time spent in conversation was 290
minutes in a sample of 20 batteries with a standard deviation of 30 minutes. Assuming that the
time is normally distributed, can it be concluded that the quality control suspicions are true at a
significance level of 1%? Construct the 99% confidence bound for the average life of electric
lamps and relate it to the result of the hypothesis test.

4. It is believed that the average level of prothrombin in a normal population is 20 mg/100 ml of
blood plasma with a standard deviation of 4 milligrams/100 ml. To verify this, a sample is taken
from 40 individuals in whom the average is 18.5 mg/100 ml. Can the hypothesis be accepted
with a significance level of 5%? Construct the 95% confidence interval for the average level of
prothrombin.

5. A company that packages peanuts states that at a maximum 6% of the peanut shells contain no
nuts. At random, 300 peanuts were selected and 21 of them were empty. With a significance
level of 1%, can the statement made by the company be accepted?

Answer 1

Note : Allowed to solve only one question in one post.

1. The life span of 100 W light bulbs manufactured by a particular company follows a normal
distribution with a standard deviation of 120 hours and its true average half-life is guaranteed
under warranty for a minimum of 800 hours.....

Hypothesis testing:

We see that there is not enough evidence to support the claim that the lot has an average half-life greater than 800. Hence the lot should be rejected for not honouring the warranty

99% confidence interval:

Yes the confidence limit reflects the same results as the hypothesis test because we see that the confidence interval (706.2872, 793.7128), does not contain 800. Indicating that there is not enough evidence to support the claim that the lot has a average half life greater than 800.