A perfume company claims that the best selling perfume contains at most 25%
alcohol. Twenty (20) bottles were selected and found to have a mean of 29.7% and
standard deviation of 4.8%. Test the claim of the perfume company at the 0.05
level of significance.
helpp
Z = .297 - .25/.048
Look up the Z score in a table in the back of your statistics text. How does it compare to .05?
Sorry, my goof.
Z = (score-mean)/SD
Z = (.25-.297)/.048
Well, well, well, looks like we have a sniffy situation here! The perfume company claims that their best selling perfume contains at most 25% alcohol. But do they really know what they're spritzing? Let's find out!
To test the claim, we need to conduct a hypothesis test. Our null hypothesis (H₀) is that the mean alcohol content is 25% or less, and the alternative hypothesis (H₁) is that the mean alcohol content is greater than 25%.
Let's do some calculations, shall we?
Step 1: Set up hypotheses:
H₀: μ ≤ 25%
H₁: μ > 25%
Step 2: Choose the significance level (α):
Given α = 0.05, which means we're willing to accept a 5% chance of making a Type I error.
Step 3: Calculate the test statistic:
We'll use the one-sample t-test since we have the sample mean and standard deviation.
Here's the formula for the test statistic: t = (x̄ - μ₀) / (s / √n)
Where x̄ is the sample mean, μ₀ is the claimed mean (25%), s is the sample standard deviation, and n is the sample size (20).
Plugging in the values, we have: t = (29.7% - 25%) / (4.8% / √20)
t = 4.7% / (4.8% / √20)
t ≈ 4.7% / 1.075
t ≈ 4.37
Step 4: Calculate the p-value:
Using the t-distribution table or a statistical calculator, we find the p-value associated with a t-value of 4.37 is pretty darn low. In fact, it's so low that we can confidently say it's less than our significance level of 0.05.
Step 5: Make a decision:
Since the p-value is less than α, we reject the null hypothesis (H₀). The evidence suggests that the mean alcohol content of the perfume is indeed greater than 25%.
Conclusion: The perfume company's claim is balloon pop! They better bottle up a new claim because their perfume actually contains more alcohol on average than they claim.
Remember, folks, when it comes to perfume claims, always sniff out the truth!
To test the claim of the perfume company, we can use a one-sample t-test.
1. Hypotheses:
- Null Hypothesis (H0): The mean alcohol content of the best selling perfume is equal to 25%.
- Alternative Hypothesis (Ha): The mean alcohol content of the best selling perfume is greater than 25%.
2. Level of Significance:
The significance level (α) is given as 0.05, which means we need to have enough evidence to reject the null hypothesis with a 5% chance of making a Type I error.
3. Test Statistic:
We will calculate the t-statistic using the sample mean, the hypothesized population mean, the sample standard deviation, and the sample size.
4. Critical Value:
Since the alternative hypothesis is one-sided, we need to find the critical value for a one-tailed test at the specified level of significance.
5. Calculate the Test Statistic:
The formula to calculate the t-statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is 29.7, the hypothesized mean is 25, the sample standard deviation is 4.8, and the sample size is 20. Let's calculate the t-statistic:
t = (29.7 - 25) / (4.8 / sqrt(20))
6. Find the Critical Value:
Using the t-table or statistical software, find the critical value for a one-tailed test at a significance level of 0.05 with degrees of freedom equal to (sample size - 1).
7. Compare the Test Statistic to the Critical Value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
By following these steps, you can perform the hypothesis test to determine whether to accept or reject the perfume company's claim.