Information given is sample mean of 12 for a sample of 26. Sample deviation is 3. Using a .02 level of significance.
H0 u< or equal to 10 and H1 u> 10
a. Is this a one or two tailed test?
b. what is the decision rule?
c. what is the value of the test statistic?
d.what is your decision regarding H0?
e. What is the p-value? and interpret it.
a. This is a one-tailed test because the alternative hypothesis (H1) states that the population mean (u) is greater than 10 (u > 10).
b. The decision rule can be determined by comparing the test statistic to the critical value(s) based on the significance level (.02). Since this is a one-tailed test and the alternative hypothesis is one-sided, we need to find the critical value for a right-tailed test. To do this, we can find the z-score associated with the desired significance level (.02) using a standard normal distribution table or a statistical calculator. The critical value will be the z-score that corresponds to a cumulative probability of .98 (1 - .02). In this case, the critical value is approximately 2.05. Therefore, if the test statistic is greater than 2.05, we would reject the null hypothesis (H0).
c. The test statistic can be calculated using the formula: test statistic = (sample mean - null hypothesis mean) / (sample deviation / sqrt(sample size)). In this case, the sample mean is 12, the null hypothesis mean is 10, the sample deviation is 3, and the sample size is 26. Plugging these values into the formula, we get: test statistic = (12 - 10) / (3 / sqrt(26)). Simplifying this, the test statistic is approximately 2.56.
d. To make a decision regarding the null hypothesis (H0), we compare the test statistic to the critical value. The test statistic (2.56) is greater than the critical value (2.05), so we would reject the null hypothesis (H0). This means that there is evidence to support the alternative hypothesis (H1) suggesting that the population mean is greater than 10.
e. The p-value can be calculated by finding the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic (2.56), assuming the null hypothesis is true. For a one-tailed test, the p-value is the area under the sampling distribution curve to the right of the observed test statistic. To find the p-value, we can use a standard normal distribution table or a statistical calculator. In this case, the p-value is approximately 0.005, which means that the probability of observing a test statistic as extreme or more extreme than 2.56, assuming the null hypothesis is true, is 0.005. It is important to note that this p-value is less than the specified significance level (.02), further supporting the decision to reject the null hypothesis.