A population =100 and sd=25. A sample (n=150) has X=102. Using 2 tails of the sampling distribution and the .05 criterion.
What is the critical value?
Is the sample in the region of rejection?
What does this indicate about the likelihood of this sample occurring in this population?
What should we conclude about the sample?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score. How does it compare to .05?
To find the critical value, we need to use the z-score formula.
The formula to calculate the z-score is:
z = (X - μ) / (σ / √n)
Given:
Population mean (μ) = 100
Population standard deviation (σ) = 25
Sample size (n) = 150
Sample mean (X) = 102
Calculating the z-score:
z = (102 - 100) / (25 / √150)
z = 2 / (25 / √150)
z ≈ 2 / 2.041
z ≈ 0.9797
The critical value is determined by the significance level (α) and the number of tails. Since we are considering two tails and the significance level is 0.05, we need to find the z-scores that correspond to a cumulative probability of 0.025 and 0.975.
Using a standard normal distribution table or calculator, the critical values for a significance level of 0.05 and a two-tailed test are approximately -1.96 and +1.96.
Since the z-score of 0.9797 is greater than +1.96, the sample is NOT in the region of rejection.
This indicates that the likelihood of this sample occurring in this population is higher than what we would consider unusual or unlikely, based on the given significance level.
Based on these results, we can conclude that there is not enough evidence to reject the null hypothesis, and the sample does not significantly differ from the population.
To find the critical value, we first need to determine the significance level (α) which is given as .05 in this case. Since we are using a two-tailed test, we need to divide the significance level by 2, resulting in α/2 = .025 for each tail.
To find the critical value, we can use a z-table or a statistical calculator. Since the population standard deviation (σ) is known, we can use the z-score formula:
Z = (X - μ) / (σ / √n)
Where:
Z is the z-score
X is the sample mean
μ is the population mean
σ is the population standard deviation
n is the sample size
In this case, the sample mean (X) is 102, the population mean (μ) is unknown, the population standard deviation (σ) = 25, and the sample size (n) = 150.
Now, we can calculate the z-score:
Z = (102 - μ) / (25 / √150)
To determine the critical value, we need to find the z-score that corresponds to the cumulative probability of 1 - α/2 (0.975 in this case) using the z-table or a statistical calculator. Let's assume the critical value is Z_critical.
If the z-score (Z) is greater than Z_critical (in either positive or negative direction), it means the sample is in the region of rejection, implying that the sample is statistically significant.
To determine if the sample is in the region of rejection, compare the calculated z-score (Z) to the critical value (Z_critical). If Z > Z_critical or Z < -Z_critical, then the sample is in the region of rejection.
The likelihood of this sample occurring in this population depends on the sample mean and the variability in the population. If the sample mean is significantly different from the population mean (i.e., in the region of rejection), it suggests that this sample is unlikely to have occurred by chance alone and there may be some underlying difference between the sample and the population.
Based on the provided information, we cannot conclude anything about the sample without knowing the population mean or conducting hypothesis testing. Further analysis is required to draw any meaningful conclusions about the sample.