Question

Read the following scenario and complete each of the problems below:

A flashlight company claims that the new bulb in its heavy duty flashlight will average 246 hours of light. A statistics student decides that he/she wants to test this claim at a 5% level of significance to determine if there is evidence to support the claim. The student randomly selects and test 15 flashlight bulbs and records how long the bulb last until it burns out. Assume the life of the bulb is normally distributed. The data is in the table below:

Trial:   1      2             3             4            5     6              7              8             9            10            11           12   13            14           15

Hrs: 246/         224/         231/       242/        234/         240/         243 /        236/         239/        255/        256/ 239/         247/         231/         253

A. Based on a histogram or box-plot of the data, what is the shape of the distribution (normal, skewed left, skewed right, uniform, etc.)? Explain why you can conduct a hypothesis test on the given data. (Compose your answer)

B. Evaluate the claim based on the following assumption: The standard deviation of the population is 7.4 hours. 1. Identify the type of test you will use to test the claim that the flashlight will average 246 hours of light. Explain your answer 2. State the null and alternate hypotheses. 3. Conduct the hypothesis test and determine the p-value. 4. State your conclusion about the claim that the flashlight will average 246 hours of light. 5. Construct a 95% confidence interval for this study.

1.

2.

3.

4.

5.

C. Evaluate the claim based on the following assumption. The population’s standard deviation is not known.

1. Identify the type of test you will use to test the claim that the flashlight will average 246 hours of light. Explain your answer.

2. State the null and alternate hypotheses.

3. Conduct the hypothesis test and determine the p-value.

4. State your conclusion about the claim that the flashlight will average 246 hours of light.

5. construct a 95% confidence interval for this study.

1.

2.

3.

4.

5.

D. Write a 200 to 400 word statement comparing and contrasting the two hypothesis test procedures and results.

Answer 1

Solution:

A] R code for plotting histogram is:

> data=c(246,224,231,242,234,240,243,236,239,255,256,239,247,231,253)
> hist(data,xlab="Lifetimes in hrs",main="Histogram of Lifetimes",col="Grey")
> library(moments)
> skewness(data)
[1] 0.05183649

Output:

                           From the histogram, it can be concluded as the shape of the distribution is skewed right. Also from Skewness =0.05183649 which indicates right skewed distribution.

We have to test if the claim of flashlight company is right or not that is we have to test that the new bulb in its heavy duty flashlight will average 246 hours of light at a 5% level of significance to determine if there is evidence to support the claim.

B]

The standard deviation of the population is 7.4 hours.

1. Identify the type of test you will use to test the claim that the flashlight will average 246 hours of light. Explain your answer

We have to use sample t test for mean with smaller sample size (15).

2. State the null and alternate hypotheses.

H0 : new bulb in its heavy duty flashlight will average 246 hours of light.

     : µ =246

V/S µ ≠ 246

And significance level=α =0.05

3. Conduct the hypothesis test and determine the p-value

R code for the test is,

> data=c(246,224,231,242,234,240,243,236,239,255,256,239,247,231,253)

> test=t.test(data,mu=246)

> test

Output:

One Sample t-test

data: data

t = -2.0612, df = 14, p-value = 0.05836

alternative hypothesis: true mean is not equal to 246

95 percent confidence interval:

235.9334 246.2000

sample estimates:

mean of x 241.066

p-value = 0.05836

4. State your conclusion about the claim that the flashlight will average 246 hours of light.

Here, p-value = 0.05836 which is greater than α =0.05

We have a weak evidence against H0.

Therefore, we accept H0 and conclude, new bulb in its heavy duty flashlight will average 246 hours of light.

5. Construct a 95% confidence interval for this study

   CI = [Mean +z²_*SE(Mean), Mean - z²_*SE(Mean)]

      95 % confidence interval is, ( 235.9334, 246.2000)

C] The procedure is same with unknown variance as we use the test statistics as,

t= (xbar−mu0)/(s/sqrt(n)) and repeat the procedure for same.