A medical researcher believes that the standard deviation of the temperatures of newborn infants is greater than 0.6 degrees. A sample of 15 infants was found to have a standard deviation of 0.8 degrees. At the 0.10 level of significance, does the evidence support the researcher’s belief? Assume that the variable is normally distributed.
To determine whether the evidence supports the researcher's belief that the standard deviation of the temperatures of newborn infants is greater than 0.6 degrees at the 0.10 level of significance, we can conduct a hypothesis test.
Let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The standard deviation of the temperatures of newborn infants is equal to or less than 0.6 degrees.
Alternative Hypothesis (Ha): The standard deviation of the temperatures of newborn infants is greater than 0.6 degrees.
The test statistic used for comparing the sample standard deviation to a hypothesized value is the chi-square statistic.
To perform the hypothesis test, we can use the following steps:
Step 1: Set the significance level (α) to 0.10.
Step 2: Calculate the test statistic:
The test statistic can be calculated using the formula:
test statistic = ((n-1) * sample standard deviation^2) / hypothesized standard deviation^2,
where n is the sample size, the sample standard deviation is 0.8 degrees, and the hypothesized standard deviation is 0.6 degrees.
Using the given information, the test statistic can be calculated as follows:
test statistic = ((15-1) * 0.8^2) / 0.6^2
Step 3: Determine the critical value:
Since we are working with a one-tailed test and a significance level of 0.10, we need to find the critical value that corresponds to the upper tail of the chi-square distribution with (n-1) degrees of freedom.
Using a chi-square table or a statistical software, the critical value for α=0.10 and (n-1=14) degrees of freedom is approximately 20.483.
Step 4: 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 5: Perform the calculation:
Substituting the values into the formula, we get:
test statistic = ((15-1) * 0.8^2) / 0.6^2
= (14 * 0.64) / 0.36
= 13.44 / 0.36
= 37.333
Step 6: Make a decision:
Since the test statistic (37.333) is greater than the critical value (20.483), we reject the null hypothesis.
Step 7: Interpret the result:
At the 0.10 level of significance, the evidence supports the researcher's belief that the standard deviation of the temperatures of newborn infants is greater than 0.6 degrees. The sample provides strong evidence against the null hypothesis.
To determine whether the evidence supports the researcher's belief, we need to perform a hypothesis test.
Step 1: State the hypotheses:
- Null Hypothesis (H0): The standard deviation of the temperatures of newborn infants is not greater than 0.6 degrees.
- Alternative Hypothesis (Ha): The standard deviation of the temperatures of newborn infants is greater than 0.6 degrees.
Step 2: Formulate an analysis plan:
- We will use a chi-square test statistic to test the null hypothesis.
- The test statistic is calculated as: chi-square = (n-1) * s^2 / σ^2, where n is the sample size, s is the sample standard deviation, and σ is the hypothesized population standard deviation.
- We will use a significance level of 0.10.
Step 3: Identify the sample data and perform calculations:
- The sample size (n) is 15.
- The sample standard deviation (s) is 0.8 degrees.
- The hypothesized population standard deviation (σ) is 0.6 degrees.
- The degrees of freedom for this test is (n - 1) = 14.
- Calculate the test statistic: chi-square = (n - 1) * (s^2) / (σ^2)
- Plug in the values: chi-square = 14 * (0.8^2) / (0.6^2)
Step 4: Determine the critical value(s):
- The critical value for a chi-square test depends on the significance level and the degrees of freedom of the test.
- With a significance level of 0.10 and 14 degrees of freedom, we can find the critical value from the chi-square distribution table or calculator.
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.
- 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.
Step 6: Interpret the decision:
- If we reject the null hypothesis, this provides evidence to support the researcher's belief that the standard deviation of the temperatures of newborn infants is greater than 0.6 degrees.