3. Let X be a random variable representing the dividend yield of Australian bank stocks. We may assume that X has a normal distribution with Now, suppose we wish to test the null hypothesis that against the alternative that using a level of significance of Ą = .05. To do so, a random sample of 16 Australian bank stocks is observed and has a sample mean of What is the p-value associated with this test?
To find the p-value associated with this test, we need to perform a hypothesis test and calculate the test statistic.
1. State the null and alternative hypotheses:
- Null hypothesis (H0): μ = 4.5%
- Alternative hypothesis (Ha): μ < 4.5%
2. Define the level of significance (α):
The significance level, α, is given as 0.05 (or 5%).
3. Calculate the test statistic:
The test statistic for this hypothesis test is a t-statistic. Since the sample size is small (n = 16) and the population standard deviation (σ) is unknown, we use a t-distribution for the test statistic.
The formula for the t-statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / √n)
Given:
Sample mean (x̄) = 4.3%
Hypothesized mean (μ) = 4.5%
Sample size (n) = 16
To calculate the sample standard deviation, we need the sample standard deviation or the sample variance. If you have the sample data, you can calculate it. If not, assume a specific value for the population standard deviation (σ) or use the sample standard deviation from a similar study.
4. Calculate degrees of freedom:
For a one-sample t-test, the degrees of freedom (df) is given by (n - 1). In this case, df = 16 - 1 = 15.
5. Look up the critical value:
Since the alternative hypothesis is one-tailed and we want to test if the mean is less than the hypothesized mean (μ < 4.5%), we find the critical value for a one-tailed test at the 5% significance level and with 15 degrees of freedom.
You can find the critical value by using a t-distribution table or a statistical calculator. The critical value will indicate the boundary below which we reject the null hypothesis.
6. Calculate the p-value:
The p-value is the probability of observing a test statistic as extreme as the one calculated (or more extreme) given that the null hypothesis is true.
To calculate the p-value, we use the t-distribution and compare the test statistic to the critical value. If the test statistic is smaller (more negative) than the critical value, the p-value is less than α, and we reject the null hypothesis.
If the test statistic is greater (less negative) than the critical value, the p-value is greater than α, and we fail to reject the null hypothesis.
The p-value can be determined by using a t-distribution table or a statistical calculator.
That's the general approach for calculating the p-value associated with this hypothesis test. Please provide the specific values for the sample standard deviation or the population standard deviation (if known), and the critical value for more accurate calculation of the p-value.