A researcher is performing a hypothesis test. The null hypothesis is that μ=10 and the alternative hypothesis is that μ<10. A sample of 18 individuals was collected and the t-based test statistic calculated from the data was -2.105. What is the approximate p-value for this hypothesis test?
To find the p-value for this hypothesis test, we need to find the area under the t-distribution curve to the left of the test statistic, t = -2.105. Since this is a one-tailed test (alternative hypothesis is μ < 10), we can use the t-distribution table, a t-distribution calculator, or statistical software to find the p-value.
Given the sample size of 18, the degrees of freedom for the t-distribution is 18 - 1 = 17.
Now, using a t-distribution calculator or statistical software, we can find the p-value:
p-value = P(T < -2.105, df = 17) ≈ 0.025
Therefore, the approximate p-value for this hypothesis test is 0.025.
To find the approximate p-value for this hypothesis test, we need to determine the probability of observing a t-value as extreme as -2.105, assuming that the null hypothesis is true.
Since the alternative hypothesis states that μ<10, this is a left-tailed test.
First, we need to find the degrees of freedom for the t-distribution used in this test. For a one-sample t-test, the degrees of freedom are calculated as n - 1, where n is the sample size. In this case, n = 18, so the degrees of freedom is 18 - 1 = 17.
Next, we use a t-table or statistical software to find the p-value corresponding to a t-value of -2.105 with 17 degrees of freedom.
Using a t-table, we find that the p-value is approximately 0.027 (assuming a significance level of 0.05). This means that the probability of observing a t-value as extreme as -2.105 under the null hypothesis is approximately 0.027.
Therefore, the approximate p-value for this hypothesis test is 0.027.
To calculate the approximate p-value for this hypothesis test, you need to know the t-distribution table or use statistical software.
Step 1: Determine the degrees of freedom.
The degrees of freedom for a t-test with a sample size of 18 individuals is given by n - 1, where n is the sample size. In this case, the degrees of freedom is 18 - 1 = 17.
Step 2: Determine the critical value for the t-test.
Since the alternative hypothesis is μ < 10, you will be conducting a one-tailed test. The critical value will depend on the significance level (α) of the test. Let's assume a significance level of α = 0.05.
Using the t-distribution table or statistical software, the critical value for a one-tailed t-test with 17 degrees of freedom and a significance level of 0.05 is approximately -1.74. This means that the rejection region for this test lies to the left of -1.74 on the t-distribution.
Step 3: Calculate the p-value.
The p-value is the probability of obtaining a test statistic as extreme as the one observed (-2.105) or more extreme, assuming the null hypothesis is true.
Since the test statistic (-2.105) falls in the rejection region to the left of the critical value (-1.74), the p-value will be less than the significance level (α = 0.05).
Using the t-distribution table or statistical software, the p-value for a one-tailed t-test with 17 degrees of freedom and a test statistic of -2.105 is approximately 0.027.
Therefore, the approximate p-value for this hypothesis test is approximately 0.027. This means there is strong evidence to reject the null hypothesis at the 0.05 level of significance, suggesting that the population mean (μ) is less than 10.