Question

Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to move forward with a purchase agreement unless it can be demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 50 lightbulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using computer software, which gave the following results.

Variable: lifetime

n: 50

Mean: 738.44

StDev: 38.20

SEMean: 5.40

t-value: -2.14

P-Value: 0.016

For all hypothesis test questions follow the following procedure:

1. State the null and alternative hypotheses and your significance level (α)

2. Choose a test statistic and check any requirements or assumptions of that statistic

3. Calculate the observed value of the test statistic and your p-value

4. Write down your decision and interpret it in the context of the original question

a) What null and alternative hypotheses were tested by the computer? Write this in symbolic form, and provide a value for μ0. Are the hypotheses that were tested correct for the context?

b) What is the conclusion if α had been set to 0.05? What if α had been set to 0.01? Express your answers in the context of the original problem.

Answer 1

(1)

H0: Null Hypothesis: 750 ( The true average life time is not smaller than what is advertised)

HA: Alternative Hypothesis: 750 ( The true average life time is smaller than what is advertised) (Claim)

(2)

(i)

Test Statistic chosen: One Sample t test

(ii) Requirements:

The Dependent Variable must be continuous. Here the Dependent Variable is lifetime of lightbulbs is continuous.

The observations are independent of one another

The Dependent Variable should be approximatelynormally distributed.

The Dependent Variable should not contain any outliers

(3)

(i)

The Test Statistic is given by:

(ii)

the observed value of the test statistic = - 2.14

df = 50 - 1 = 49

By Technology,

p - value = 0.016

(4)

Since p - value =0.016 is less than = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the true average life time is smaller than what is advertised. So, it can be demonstrated that the true average lifetime is smaller than what is advertised. Hence customer should not decide to move forward with a purchase agreement.

(a)

The null and alternative hypotheses that were tested by the computer are as follows:

H0: Null Hypothesis: 750 ( The true average life time is not smaller than what is advertised)

HA: Alternative Hypothesis: 750 ( The true average life time is smaller than what is advertised) (Claim)

The hypotheses that were tested are correct for the context.

(b)

(i)

If α had been set to 0.05 :

Since p - value =0.016 is less than = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the true average life time is smaller than what is advertised. So, it can be demonstrated that the true average lifetime is smaller than what is advertised. Hence customer should not decide to move forward with a purchase agreement.

(ii)

If α had been set to 0.01 :

Since p - value =0.016 is greater than = 0.01, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not support the claim that the true average life time is smaller than what is advertised. So, it cannot be demonstrated that the true average lifetime is smaller than what is advertised. Hence customer should decide to move forward with a purchase agreement.