Question

Test scores on a Spanish language proficiency exam given to entering first year students at a certain university have a mean score of 78 with a standard deviation of 9 points (out of 100 points possible).

3.b

Suppose a random sample of 50 first year students who took this Spanish placement exam will be selected, each of their scores are recorded, and the sample mean score out of 100 points will be computed. What can be said about the probability that the sample mean Spanish proficiency score will be at least 80?

• 0.5871

• 0.0582

• 0.4129

• 0.9418

• It cannot be determined accurately from the information given.

3.c

Suppose computing the data for 50 students took too much time, and a random sample of a smaller size was taken such that the Central Limit Theorem would still apply.
Describe how this change in sample size would have affected the following, if at all:

The mean of the distribution for the sample mean Spanish placement exam score for a first year student at this university would ___________.

• Increase

• Decrease

• Remain unchanged

• Insufficient information to tell what would happen

3.d

Suppose computing the data for 50 students took too much time, and a random sample of a smaller size was taken such that the Central Limit Theorem would still apply.
Describe how this change in sample size would have affected the following, if at all:

The standard deviation of the distribution for the sample mean Spanish placement exam score for a first year student at this university would ___________.

• Increase

• Decrease

• Remain unchanged

• Insufficient information to tell what would happen

3.e

Suppose computing the data for 50 students took too much time, and a random sample of a smaller size was taken such that the Central Limit Theorem would still apply.
Describe how this change in sample size would have affected the following, if at all:

The probability that the sample mean Spanish proficiency score will be at least 80 would _____________.

• Increase

• Decrease

• Remain unchanged

• Insufficient information to tell what would happen

Answer 1

Answer 2

3.b: To answer this question, we can use the central limit theorem. Since the sample size is large (n=50), we can assume that the distribution of sample means will be approximately normal, with a mean of 78 and a standard deviation of 9/sqrt(50) = 1.27. To find the probability that the sample mean score will be at least 80, we can standardize the sample mean:

z = (80 - 78) / 1.27 = 1.57

Using a standard normal distribution table or calculator, we find that the probability of a z-score being greater than 1.57 is approximately 0.0582. Therefore, the answer is option B: 0.0582.

3.c: According to the central limit theorem, as the sample size increases, the distribution of sample means becomes more normal, and the mean of the distribution approaches the population mean. Therefore, if a smaller sample is taken, the mean of the distribution for the sample mean Spanish placement exam score would remain unchanged.

3.d: The standard deviation of the distribution for the sample mean Spanish placement exam score is given by the population standard deviation divided by the square root of the sample size. Therefore, as the sample size decreases, the standard deviation of the distribution for the sample mean Spanish placement exam score would increase. Thus, the answer is option A: increase.

3.e: The probability that the sample mean Spanish proficiency score will be at least 80 depends on the mean and standard deviation of the distribution for the sample mean, which in turn depend on the sample size. As the sample size decreases, the standard deviation of the distribution for the sample mean increases, which means that the probability of obtaining a sample mean of at least 80 decreases. Therefore, the answer is option B: decrease.

Answer 3

3.a. The probability of a student randomly selected from the university having a Spanish proficiency score greater than 80 can't be determined from the information given.

3.b. To solve this question, we need to calculate the z-score and find the probability using a z-table or calculator. The z-score for a sample mean of 80 with a sample size of 50 can be calculated as follows:

z = (x̄ - μ) / (σ / √n) where x̄ = sample mean = 80 μ = population mean = 78 σ = population standard deviation = 9 n = sample size = 50

z = (80 - 78) / (9 / √50) = 2.22

Looking up the probability from a z-table or using a calculator, we get a probability of 0.0139. Therefore, the probability that the sample mean Spanish proficiency score will be at least 80 is approximately 0.0139 or 1.39%.

3.c. According to the Central Limit Theorem, as long as the sample size is sufficiently large (usually n ≥ 30), the distribution of the sample means will be approximately normal, regardless of the shape of the population distribution. As a result, the mean of the distribution for the sample mean Spanish placement exam score for a first year student at this university would remain unchanged, assuming the sample size is still sufficiently large.

3.d. The standard deviation of the distribution for the sample mean Spanish placement exam score for a first year student at this university would decrease as the sample size decreases. This is because as the sample size increases, the variability of the sample means decreases, and as the sample size decreases, the variability of the sample means increases.

3.e. The probability that the sample mean Spanish proficiency score will be at least 80 would be affected by the change in sample size, but we cannot determine exactly how it would be affected without knowing the new sample size. However, as the sample size decreases, the variability of the sample means increases, which means that the probability of obtaining a sample mean of 80 or higher would decrease.