Hi there, I am doing my statistics homework and am having trouble with a few problems. I sincerely appreciate any help I receive! :)
A manufacturer claims that the mean lifetime of its lithium batteries if 1200 hours. A homeowner selects 25 of there batteries and finds the mean lifetime to be 1180 hours with a standard deviation of 80 hours. Test the manufacturers claim. Use (alpha= .05), test the claim
To test the manufacturer's claim, we can use a hypothesis test. Here's a step-by-step guide on how to approach this problem:
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) is the statement we want to test. In this case, the manufacturer's claim is that the mean lifetime of the batteries is 1200 hours.
So, the null hypothesis can be stated as:
H0: The mean lifetime of the batteries is 1200 hours.
The alternative hypothesis (Ha) is the opposite of the null hypothesis. It represents the possibility that the claim is not true. Here, the alternative hypothesis can be stated as:
Ha: The mean lifetime of the batteries is not 1200 hours.
Step 2: Determine the significance level (alpha).
The significance level (alpha) is the maximum probability you are willing to accept for rejecting the null hypothesis when it is true. In this case, the significance level is given as 0.05 or 5%.
Step 3: Identify the test statistic.
Since we have the sample mean and standard deviation, we can use the t-test statistic. The formula is given as:
t = (sample mean - population mean) / (standard deviation / √sample size)
Step 4: Calculate the test statistic.
Plugging in the given values, we have:
sample mean (x̄) = 1180
population mean (μ) = 1200
standard deviation (σ) = 80
sample size (n) = 25
t = (1180 - 1200) / (80 / √25)
t = -20 / (80 / 5)
t = -20 / 16
t = -1.25
Step 5: Determine the critical value(s).
To determine the critical value(s), we need to refer to the t-distribution table. With a two-tailed test at a 5% significance level and 24 degrees of freedom (sample size minus 1), the critical values are approximately ±2.064.
Step 6: Compare the test statistic with the critical value(s).
Since -1.25 falls between -2.064 and 2.064, it does not fall in the critical region. Therefore, we fail to reject the null hypothesis.
Step 7: State the conclusion.
Based on the test, there is not enough evidence to reject the manufacturer's claim that the mean lifetime of the batteries is 1200 hours at a 5% significance level.
Note: It's important to remember that failing to reject the null hypothesis does not imply that the claim is true, but rather that there is not sufficient evidence to suggest otherwise.
To test the manufacturer's claim, we can use a hypothesis test. Here are the steps to follow:
Step 1: Set up the hypothesis:
- Null hypothesis (H0): The mean lifetime of the lithium batteries is equal to 1200 hours.
- Alternative hypothesis (Ha): The mean lifetime of the lithium batteries is different from 1200 hours.
Step 2: Determine the significance level:
The significance level (alpha) is given as 0.05 in this case. This means that we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true).
Step 3: Conduct the test using the sample data:
Calculate the test statistic, which is a measure of how far the sample mean is from the claimed mean, assuming the null hypothesis is true. In this case, we can use the t-test because the sample size is less than 30, and the population standard deviation is unknown.
The formula for the t-test statistic is: t = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean (1180 hours),
μ is the claimed population mean (1200 hours),
s is the sample standard deviation (80 hours),
and n is the sample size (25).
Step 4: Determine the critical region(s):
Since the alternative hypothesis is two-sided (the mean lifetime could be either greater or less than 1200 hours), we need to find the critical values in both tails of the distribution.
To find the critical values, we need to consult the t-distribution table or use statistical software.
Step 5: Calculate the p-value:
The p-value measures the probability of observing a test statistic as extreme as the one calculated from the sample data, assuming the null hypothesis is true. You can determine the p-value using a t-distribution table or software.
Step 6: Make a decision and interpret the results:
- If the p-value is less than the significance level (alpha), reject the null hypothesis. This means there is enough evidence to claim that the mean lifetime of the lithium batteries is different from 1200 hours.
- If the p-value is greater than the significance level (alpha), fail to reject the null hypothesis. This means there is not enough evidence to claim that the mean lifetime of the lithium batteries is different from 1200 hours.
Remember to cross-check your calculations and assumptions, and use a calculator or statistical software to perform the calculations accurately.