In the population of typical college students, _ 5 75 on a statistics final exam _X 5 6.42 . For 25 students who studied statistics using a new technique, X 5 72.1. Using two tails of the sampling distribution and the .05 criterion: (a) What is the critical value? (b) Is this sample in the region of rejection? How do you know? (c) Should we conclude that the sample represents the population of typical students? (d) Why?

Why are you substituting 5 for =?

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.025 because it is two-tailed) and its Z score. Insert Z first as positive and later as negative into above equation.

Calculate and come to your own conclusions.

(a) Ah, the critical value! It's like the bouncer at the statistical party. To find it, we need to know the degrees of freedom. Do you know how many degrees of freedom we have in this case?

(b) Now, let's see if our sample is in the region of rejection. If our test statistic falls in that region, then it's time for our sample to pack their bags and leave the party. Did we get any info on where the test statistic falls?

(c) Ah, the classic question of whether the sample represents the population. It's like wondering if a clown car represents all cars. To make a conclusion, we need to consider the p-value. Do we have any information about the p-value?

(d) And why should we make this conclusion? Well, that's where the p-value comes in. If it's smaller than our significance level (0.05), then we have enough evidence to reject the null hypothesis and say that the sample is different from the population. But without knowing the p-value, we can't say for sure. Do we have any insights into the p-value?

To answer these questions, we need to perform a hypothesis test. Let's go through each step:

Step 1: Define the hypotheses:
Null hypothesis (H0): The sample represents the population of typical college students. (μ = 75)
Alternative hypothesis (Ha): The sample does not represent the population of typical college students. (μ ≠ 75)

Step 2: Determine the significance level:
The significance level is given as α = 0.05.

Now let's proceed to answer each question:

(a) What is the critical value?
To determine the critical value, we need to look up the z-value associated with a significance level of 0.05 and a two-tailed test. Checking a standard normal distribution table, the critical value corresponds to ±1.96.

(b) Is this sample in the region of rejection? How do you know?
We can determine if the sample is in the region of rejection by calculating the test statistic and comparing it to the critical value. The formula to calculate the z-test statistic is:

z = (X - μ) / (σ / sqrt(n))

Given that X (sample mean) is 72.1, μ (population mean) is 75, σ (population standard deviation) is 6.42, and n (sample size) is 25:

z = (72.1 - 75) / (6.42 / sqrt(25))
z = -2.9

Since the absolute value of the z-test statistic (-2.9) is greater than the critical value (1.96), we can conclude that the sample is in the region of rejection.

(c) Should we conclude that the sample represents the population of typical students?
Based on the result in (b), we reject the null hypothesis that the sample represents the population of typical college students. Therefore, we should not conclude that the sample represents the population of typical students.

(d) Why?
The sample falls in the region of rejection, suggesting that it is unlikely to be a representative sample of the population. It indicates that the sample mean of 72.1 is significantly different from the population mean of 75. To draw stronger conclusions, further investigation or a larger sample may be necessary.

Note: The calculation assumes that the population standard deviation is known. If the population standard deviation is unknown and must be estimated from the sample, a t-test should be used instead.

To answer these questions, we need to understand the process of hypothesis testing and how to calculate the critical value and the test statistic.

(a) Calculating the critical value:
To find the critical value, we need to determine the significance level or the alpha level. In this case, the significance level is 0.05 since it is mentioned in the question. The significance level, denoted by α, determines how unlikely the result must be in order to reject the null hypothesis.

In a two-tailed test, the total significance level is divided equally between the two tails. So, we need to find the critical value for the α/2 from the z-table. In this case, α/2 = 0.05/2 = 0.025.

Using a standard normal distribution table (z-table), we can find the z-value corresponding to a cumulative probability of 0.975 (1 - α/2). The critical value will be the z-score corresponding to this probability.

(b) Determining if the sample is in the region of rejection:
We can determine if the sample is in the region of rejection by comparing the test statistic with the critical value obtained in (a).

In this case, we compare the sample mean (X = 72.1) with the population mean (μ = 65). Since the information about the population standard deviation is not provided, we'll assume it to be the same as the typical college students' standard deviation (σ = 6.42).

Using the formula for calculating the test statistic for a single sample t-test:

t = (X - μ) / (σ / sqrt(n))

Where:
- X is the sample mean,
- μ is the population mean,
- σ is the population standard deviation (or assumed to be the population standard deviation),
- n is the sample size.

(c) Determining if we can conclude that the sample represents the population of typical students:
To determine if we can conclude that the sample represents the population of typical students, we need to consider the significance level and the p-value.

The p-value is the probability of observing a test statistic as extreme as the one calculated or more extreme, assuming the null hypothesis is true. If the p-value is less than the significance level (α), we can reject the null hypothesis in favor of the alternative hypothesis.

(d) Explaining why we should conclude or not conclude:
To reach a conclusion, we compare the p-value with the significance level (α) to decide whether to reject the null hypothesis or fail to reject it.

Now, let's calculate the critical value, test statistic, and p-value to answer the questions:

(a) Finding the critical value:
- Look up the cumulative probability of 0.975 in the z-table, which is approximately 1.96.
- The critical value, denoted as z_crit, will be positive and negative 1.96.

(b) Determining if the sample is in the region of rejection:
- Calculate the test statistic using the formula mentioned earlier:
t = (72.1 - 65) / (6.42 / sqrt(25))
= 6.10

- Since the sample is a two-tailed test, compare the absolute value of the test statistic (|t|) with the absolute value of the critical value (|z_crit|). If |t| > |z_crit|, the sample is in the region of rejection.

(c) Deciding if we can conclude that the sample represents the population of typical students:
- Calculate the p-value using the test statistic:
p-value = P(|t| > 6.10)

(d) Explaining the conclusion:
- If the p-value is less than the significance level (α = 0.05), we can reject the null hypothesis and conclude that the sample represents the population of typical students. If the p-value is greater than or equal to α, we fail to reject the null hypothesis.

To complete this answer, we need the p-value corresponding to the test statistic 6.10. It can be obtained using a t-table or a statistical software program. However, the exact p-value calculation is beyond the scope of this explanation.