If it
is claimed that the contents
of soaps
boxes sold
weigh on average at least 20 ounces.
The distribution
of weight is known to be normal;
We
assume \sigma = 0.5σ=0.5 is known
n=30
sample
mean
weight =
18.5 ounces.
Test
at
the 10% significance level (\alpha = .10)(α=.10) the null
hypothesis that the population mean weight is at least 20
ounces.
To test the null hypothesis that the population mean weight is at least 20 ounces, we can use a one-sample z-test.
Here are the steps to perform the test:
Step 1: Define the null and alternative hypotheses:
The null hypothesis (H0) is that the population mean weight is equal to or less than 20 ounces.
The alternative hypothesis (H1) is that the population mean weight is greater than 20 ounces.
Step 2: Determine the significance level (alpha):
In this case, the significance level is given as 0.10 (or 10%).
Step 3: Calculate the test statistic:
The test statistic for a one-sample z-test is given by the formula:
z = (x̄ - μ) / (σ/√n)
where x̄ is the sample mean, μ is the hypothesized population mean, σ is the known standard deviation, and n is the sample size.
In this case, the sample mean weight is 18.5 ounces, the hypothesized population mean is 20 ounces, the known standard deviation is 0.5 ounces, and the sample size is 30.
Plugging these values into the formula, we get:
z = (18.5 - 20) / (0.5 / √30)
Step 4: Calculate the critical value:
Since we are testing the null hypothesis against the alternative hypothesis that the population mean is greater than 20 ounces, we need to look up the critical value in the right-tail of the standard normal distribution.
For a significance level of 0.10, the critical z-value is approximately 1.28.
Step 5: Compare the test statistic with the critical value:
If the test statistic calculated in Step 3 is greater than the critical value obtained in Step 4, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Step 6: Make a conclusion:
Based on the comparison in Step 5, if the test statistic is greater than the critical value, we can conclude that there is sufficient evidence to reject the null hypothesis and support the alternative hypothesis that the population mean weight is greater than 20 ounces. Otherwise, if the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis and do not have sufficient evidence to support the claim.
Note: Please calculate the z-value using the given values in Step 3 to finalize the test.