A manufacturer wants to increase the absorption capacity of a sponge. Based on past data, the average sponge could absorb 3.5 ounces. After the redesign, the absorption amounts of a sample of sponges were (in ounces): 4.1, 3.7, 3.3, 3.5, 3.8, 3.9, 3.6, 3.8, 4.0, and 3.9. What is the decision rule at the 0.01 level of significance to test if the new design increased the absorption amount of the sponge
Do not reject the null hypothesis if the T value is less than 2.812
To test if the new design increased the absorption amount of the sponge, we can use a one-sample t-test.
Step 1: State the hypothesis
The null hypothesis (H0) assumes that there is no significant increase in the absorption amount of the sponge, while the alternative hypothesis (H1) assumes that there is a significant increase.
H0: The average absorption amount of the sponge is equal to 3.5 ounces.
H1: The average absorption amount of the sponge is greater than 3.5 ounces.
Step 2: Select the significance level
In this case, the significance level is given as 0.01.
Step 3: Compute the test statistic
We'll use the T-test statistic to determine whether the sample mean is significantly different from the hypothesized mean. The formula for the T-test statistic is:
t = (x̄ - μ) / (s / √n)
Where:
x̄ represents the sample mean,
μ represents the hypothesized mean,
s represents the sample standard deviation, and
n represents the sample size.
Step 4: Formulate the decision rule
For a one-tailed test at the 0.01 significance level, we'll reject the null hypothesis if the t-test statistic is greater than the critical value.
Step 5: Calculate the test statistic and compare to the critical value
Given data: 4.1, 3.7, 3.3, 3.5, 3.8, 3.9, 3.6, 3.8, 4.0, and 3.9
Sample mean (x̄): (4.1 + 3.7 + 3.3 + 3.5 + 3.8 + 3.9 + 3.6 + 3.8 + 4.0 + 3.9) / 10 = 3.82
Sample standard deviation (s): √ [ ((4.1 - 3.82)^2 + (3.7 - 3.82)^2 + ... + (3.9 - 3.82)^2) / (10 - 1) ] = 0.27
Hypothesized mean (μ): 3.5
Sample size (n): 10
t = (x̄ - μ) / (s / √n) = (3.82 - 3.5) / (0.27 / √10) = 2.97
Step 6: Compare the test statistic to the critical value
We'll use a one-tailed t-test table or a statistical software to find the critical value corresponding to a significance level of 0.01 and 9 degrees of freedom. Let's assume the critical value is 2.821. Since the test statistic (2.97) is greater than the critical value (2.821), we can reject the null hypothesis.
Step 7: Make a decision
Based on the calculated test statistic and the critical value, we can reject the null hypothesis. This suggests that the new sponge design has indeed increased the absorption amount.
In summary, the decision rule at the 0.01 level of significance is to reject the null hypothesis if the calculated t-test statistic is greater than the critical value.
To test if the new design increased the absorption amount of the sponge, we can perform a hypothesis test using the data provided.
Here are the steps to conduct the hypothesis test:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The new design did not increase the absorption amount of the sponge.
- Alternative hypothesis (Ha): The new design increased the absorption amount of the sponge.
Step 2: Determine the test statistic:
Since we have a small sample size (n < 30) and the population standard deviation is unknown, we can use the t-distribution for this test. We will calculate the t-statistic using the sample mean, sample standard deviation, and the sample size.
Step 3: Set the significance level (α):
The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is true. In this case, the significance level required is 0.01, which means we are willing to accept a 1% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 4: Calculate the test statistic:
- Compute the sample mean (x̄) using the provided absorption amounts.
- Compute the sample standard deviation (s) using the provided absorption amounts.
- Determine the sample size (n).
Step 5: Determine the critical value:
- Look up the critical value for a two-tailed t-test with n-1 degrees of freedom and a significance level of 0.01. The degrees of freedom is equal to the sample size minus one (df = n - 1).
Step 6: Decision rule:
- If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis.
- If the absolute value of the test statistic is less than or equal to the critical value, fail to reject the null hypothesis.
Step 7: Compute the test statistic:
- Calculate the t-statistic using the formula: t = (x̄ - μ) / (s / √n), where μ is the assumed population mean (3.5 ounces in this case).
Step 8: Make a decision:
- Compare the calculated t-value with the critical value determined earlier. If the calculated t-value falls in the critical region, reject the null hypothesis. If it falls within the non-critical region, fail to reject the null hypothesis.
Please note that the calculations for the test statistics and critical values are not shown in this explanation. They can be determined using statistical software, a t-distribution table, or a calculator with statistical functions.