Question

A manufacturer of light bulbs has a bulb that, on the average, bums for 2000 hours. The research deparmment has developed a new bulb that it claims outlasts the older type. A test of 52 new bulbs yields a mean of 2011 hours with a standurd deviation of 35.6 . Ai the 1% significance level, test whether the new balb is longer buming than the old, on average. Periom the corresponding hypothesis test using any approach. A. Define both lypotheses. B. Calculate the teat statistic. C. Calculate the p-value of use the classical approsich. D. Do we reject the H. Explain why. E. Write your conclusion in context with the problem

Answer 1

To determine if the new bulb lasts longer, we define null and alternative hypotheses, calculate the test statistic, find the corresponding p-value, and then decide whether to reject the null hypothesis based on the p-value relative to the 1% significance level.

Step-by-step explanation:

Hypothesis Testing for a New Light Bulb's Longevity

First, we'll define the hypotheses for testing whether the new bulb has a longer average burning time than the old one, which has an average of 2000 hours:

• Null Hypothesis (H0): The mean burning time of the new bulb ≤ 2000 hours (the new bulb does not outlast the older one on average).
• Alternative Hypothesis (Ha): The mean burning time of the new bulb > 2000 hours (the new bulb outlasts the older one on average).

Next, we calculate the test statistic using the sample mean, the population mean, and the standard deviation:

1. Compute the standard error (SE) = σ/√(n) = 35.6/√(52).
2. Calculate the z-score = (2011 - 2000) / SE.

Then, we find the p-value corresponding to the z-score in the standard normal distribution. This is a one-tailed test because the alternative hypothesis indicates a direction of 'greater than.'

If the p-value is less than 0.01 (the level of significance), we reject H0, indicating that the new bulb is statistically significantly longer burning than the old one. Otherwise, we fail to reject H0.

The conclusion will state whether there is sufficient evidence at the 1% significance level to support the research department's claim that the new bulb outlasts the older type.