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 the standard deviation of salt in their meat is .5. An SRS of 18 samples shows the standard deviation (s)= .62
(alpha = .05)
Is this evidence that the standard deviation is > .5?
To determine if there is evidence that the standard deviation of the salt in the meat is greater than 0.5, we can conduct a hypothesis test.
Step 1: State the hypotheses:
- Null hypothesis (H₀): The standard deviation of salt in the meat is equal to 0.5 (σ = 0.5).
- Alternative hypothesis (H₁): The standard deviation of salt in the meat is greater than 0.5 (σ > 0.5).
Step 2: Set the significance level:
The given significance level in this case is α = 0.05.
Step 3: Compute the test statistic:
In this case, we will compare the sample standard deviation (s) to the hypothesized standard deviation (σ₀) using the Chi-Square test statistic, given by the formula:
χ² = (n - 1) * (s / σ₀)²
where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized standard deviation.
Given:
n = 18 (sample size)
s = 0.62 (sample standard deviation)
σ₀ = 0.5 (hypothesized standard deviation)
Calculate the test statistic:
χ² = (18 - 1) * (0.62 / 0.5)²
Step 4: Determine the critical value:
Since the alternative hypothesis is looking for evidence of the standard deviation being greater than 0.5, we will use the right-tailed test. Using the chi-square distribution table or calculator, find the critical value at an alpha level of 0.05 and the degrees of freedom (n - 1).
Step 5: Make a decision:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value:
If using software or a chi-square calculator, we can calculate the p-value associated with the test statistic. The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed test statistic, assuming the null hypothesis is true.
Step 7: Make a conclusion:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis and do not have sufficient evidence to support the alternative hypothesis.
By following these steps, you should be able to determine if there is evidence that the standard deviation of salt in the meat is greater than 0.5 based on the given sample data.