A manufacturer is interested in determining whether it can claim that the boxes of detergent it sells contain, on average, more than 500 grams of detergent. The firm selects a random sample of 100 boxes and records the amount of detergent (in grams) in each box. The data are provided in the file P09_02.xlsx.
a. Identify the null and alternative hypotheses for this situation
b. Is there statistical support for the manufacturer’s claim? Explain.
Ho: mean > 500
Ha: mean ≤ 500
Data missing.
a. The null hypothesis (H0) is that the boxes of detergent contain, on average, 500 grams of detergent. The alternative hypothesis (Ha) is that the boxes of detergent contain, on average, more than 500 grams of detergent.
b. To determine if there is statistical support for the manufacturer's claim, we can perform a one-sample t-test. Here, we compare the average amount of detergent in the sample to the claimed value of 500 grams.
To achieve this, we need to calculate the sample mean and standard deviation using the provided data in the P09_02.xlsx file. Then, we can conduct the one-sample t-test using the following steps:
1. Calculate the sample mean (x̄) and the sample standard deviation (s).
2. Set the significance level (α) to determine the critical value and accept or reject the null hypothesis accordingly. Let's assume α = 0.05.
3. Determine the critical value or p-value for the t-statistic using the t-distribution table or statistical software.
4. Calculate the t-statistic using the formula t = (x̄ - μ) / (s / (√n)), where μ is the population mean (claimed value) and n is the sample size.
5. Compare the calculated t-statistic with the critical value. If the calculated t-statistic is greater than the critical value (or if the p-value is less than α), we reject the null hypothesis and accept the alternative hypothesis.
By performing these steps, we can determine if there is statistical support for the manufacturer's claim.
To answer these questions, you would need to perform a hypothesis test. Here's how you can go about it:
a. Identifying the null and alternative hypotheses:
The null hypothesis (H₀) represents the claim that the manufacturer wants to test. In this case, the null hypothesis would be that the boxes of detergent contain, on average, 500 grams of detergent. The alternative hypothesis (H₁) would be the opposite of the null hypothesis, stating that the boxes of detergent contain, on average, more than 500 grams of detergent.
So, the null hypothesis (H₀) is: The average amount of detergent in the boxes is 500 grams.
And the alternative hypothesis (H₁) is: The average amount of detergent in the boxes is greater than 500 grams.
b. Determining statistical support:
To determine if there is statistical support for the manufacturer's claim, you would need to conduct a hypothesis test using the given data. Here are the steps to perform the test:
1. Calculate the sample mean and sample standard deviation of the 100 boxes' detergent amount.
2. Set a significance level (alpha) for the test. The significance level determines the probability of rejecting the null hypothesis when it is actually true. A common choice is 0.05.
3. Use the sample data to calculate the test statistic. In this case, you would use the t-statistic since the population standard deviation is not known and it's a relatively small sample size (n < 30).
4. Determine the critical value (or p-value) based on the chosen significance level and the degrees of freedom of the t-distribution.
5. Compare the test statistic with the critical value (or p-value) to make a decision. If the test statistic falls in the rejection region (beyond the critical value) or the p-value is less than the significance level, then you can reject the null hypothesis in favor of the alternative hypothesis. Otherwise, if the test statistic does not fall in the rejection region or the p-value is greater than the significance level, you fail to reject the null hypothesis.
Based on the outcome of the hypothesis test, you can determine whether there is statistical support for the manufacturer's claim that the boxes of detergent contain, on average, more than 500 grams.