The mean of a population of raw scores is 33 1_X 5 122 . Use the criterion of .05 and the upper tail of the sampling distribution to test whether a sample with X 5 36.8 1N 5 302 represents this population. (a) What is the critical value? (b) Is the sample in the region of rejection? How do you know? (c) What does this indicate about the likelihood of this sample occurring in this population? (d) What should we conclude about the sample?
To test whether a sample with a mean of 36.8 and a sample size of 302 represents a population with a mean of 33, we need to perform a hypothesis test.
Step 1: State the hypotheses.
The null hypothesis (H0): The sample does represent the population (μ = 33).
The alternative hypothesis (Ha): The sample does not represent the population (μ ≠ 33).
Step 2: Determine the significance level and test statistic.
The significance level (α) is given as 0.05. Since we are testing the upper tail of the distribution, the critical value will be based on the upper tail of the sampling distribution.
Step 3: Calculate the critical value.
Since the significance level is 0.05, we need to find the z-value that corresponds to an area of 0.05 in the upper tail of the standard normal distribution. By referring to a standard normal distribution table, we find that the critical value is approximately 1.645.
(a) The critical value is approximately 1.645.
Step 4: Calculate the test statistic.
To calculate the test statistic, we need to calculate the z-score using the formula:
z = (sample mean - population mean) / (population standard deviation / √sample size)
Given:
Sample mean (X̄) = 36.8
Population mean (μ) = 33
Population standard deviation (σ) = unknown (not provided)
Sample size (n) = 302
Unfortunately, we do not have the population standard deviation, which is needed to calculate the z-score. Without this information, we cannot proceed with the hypothesis test.
(b) Since we do not have the information to calculate the test statistic, we cannot determine whether the sample is in the region of rejection.
(c) Without the ability to calculate the test statistic, we cannot assess the likelihood of this sample occurring in this population.
(d) In conclusion, due to the lack of information about the population standard deviation, we cannot make any conclusions about the sample and whether it represents the population.
To determine the critical value and answer the subsequent questions, we need to perform a hypothesis test. In this case, we are testing whether the sample with a mean of 36.8 represents a population with a mean of 33.
(a) To find the critical value, we need to determine the rejection region based on the given criterion of .05 and the upper tail of the sampling distribution. Since the criterion is .05, we are looking for the critical region that contains 5% of the area under the curve in the upper tail. We can use a Z-table or a statistical calculator to find the critical value associated with this significance level.
(b) To determine if the sample is in the region of rejection, we compare the test statistic (sample mean) to the critical value. If the test statistic falls in the critical region, we reject the null hypothesis. If it falls outside the critical region, we fail to reject the null hypothesis. In this case, if the sample mean of 36.8 is greater than the critical value, it will be in the region of rejection.
(c) If the sample mean falls within the region of rejection, it indicates that the likelihood of getting a sample mean of 36.8 (or more extreme) from the population with a mean of 33 is very low. In other words, obtaining such a sample is highly unlikely if the population mean is truly 33.
(d) Based on the results, if the sample mean falls within the region of rejection, we reject the null hypothesis. Therefore, we can conclude that the sample does not represent the population with a mean of 33.
Please note that I wasn't able to fully understand the notation used in your question ("1_X", "5", "1N", "302"). If you can provide additional information or clarify the notation, I would be happy to assist you further.