1) A light bulb producing company states that its lights will last an average of 1200 hours with a standard deviation of 200 hours. A sample of 100 light bulbs from the company were tested and the researcher found that the average life of each light bulb was 1050 hours. At a 95% confidence level, determine whether these light bulbs are in compliance with the company's claim.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score for your alpha level.
To determine whether the light bulbs are in compliance with the company's claim, we can perform a hypothesis test using the given data.
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The average life of the light bulbs is equal to 1200 hours.
- Alternative hypothesis (Ha): The average life of the light bulbs is not equal to 1200 hours.
Step 2: Determine the significance level (α):
The significance level, denoted as α, is the probability of rejecting the null hypothesis when it's true. In this case, the significance level is not explicitly mentioned, so we'll use the commonly used value of α = 0.05 (or 5%).
Step 3: Calculate the test statistic:
To calculate the test statistic, we'll use the formula for a one-sample z-test:
z = (sample mean - population mean) / (standard deviation / √sample size)
In this case:
Sample mean (x̄) = 1050 hours
Population mean (μ) = 1200 hours
Standard deviation (σ) = 200 hours
Sample size (n) = 100 bulbs
Plugging in the values, we get:
z = (1050 - 1200) / (200 / √100)
z = (1050 - 1200) / (200 / 10)
z = -150 / 20
z = -7.5
Step 4: Determine the critical value(s):
Since we have a two-tailed test (Ha: average life ≠ 1200 hours), we need to find the critical value(s) for a 95% confidence level. The critical value (z) can be obtained from the standard normal distribution table or a statistical calculator. For a 95% confidence level, the critical value is approximately ±1.96.
Step 5: Compare the test statistic with the critical value:
Since the calculated test statistic (-7.5) falls outside the critical value range of ±1.96, we can reject the null hypothesis (H0).
Step 6: Make the conclusion:
Based on the hypothesis test, at a 95% confidence level, we have sufficient evidence to conclude that the average life of the light bulbs is not equal to 1200 hours. Therefore, the light bulbs are not in compliance with the company's claim.
Note: The conclusion is based on the assumption that the sample is representative of the entire population of light bulbs produced by the company.