Variability in Estimates

Question 4.3 - College Credits: A college counselor is interested in estimating how many credits a student typically enrolls in each semester. The counselor decides to randomly sample 100 students by using the registrar’s database of students. The histogram shows the distribution of the number of credits taken by these students. Sample statistics for this distibution are also provided.

What we know from the graph: Min: 8 Q1: 13 Median: 14 Mean: 13.65 SD: 1.91 Q3: 15 Max: 18

a. What is the point estimate for the average number of credits taken per semester by students at this college? What about the median?

Answer: From the information above (and next to the graph in the book), we know that the average, or mean, number of credits taken is 13.65. The median is 14.

b. What is the point estimate for the standard deviation of the number of credits taken per semester by students at this college? What about the IQR?

Answer: The point estimate for our sample of 100 students is 1.91 (or the SD listed above). The IRQ: Q3 - Q1

Q3 <- 15
Q1 <- 13
Q3 - Q1
## [1] 2

The IQR is 2

c. Is a load of 16 credits unusually high for this college? What about 18 credits? Explain your reasoning. Hint: Observations farther than two standard deviations from the mean are usually considered to be unusual.

Answer: First, I’ll figure out the Z score of 16 and 18 credits. The formula is: Z = observation - mean/SD

mn <- 13.65
sd <- 1.91
x1 <- 16
x2 <- 18

(x1 - mn) / sd
## [1] 1.230366
(x2 - mn) / sd
## [1] 2.277487

Based on the information above, we see that 16 falls within 2 SD of the mean, which makes 16 credits a normal course load but 18 credits would be unusual.

d. The college counselor takes another random sample of 100 students and this time finds a sample mean of 14.02 units. Should she be surprised that this sample statistic is slightly different than the one from the original sample? Explain your reasoning.

Answer: No, there is always natural variability in the sample statistic and 14.02 still falls within the 2 standard deviation of the mean. We would be more surprised if the data was exactly the same!

e. The sample means given above are point estimates for the mean number of credits taken by all students at that college. What measures do we use to quantify the variability of this estimate?

Answer: We use the standard error to quantify the variability. A sample mean deviates from the actual mean of a population; this deviation is the standard error. We can calculate this by:

sd <- 1.91
n <- sqrt(100)
sd/n
## [1] 0.191

The standard error of mean is 0.191