Question

In Lesson Seven you've learned to convert raw scores to standard scores and used the empirical rule to determine probabilities associated with those standard scores.

Random Decimal Fraction Generator

Here are your random numbers:

0.1613
0.8590
0.7911

1. Use the random decimal fraction generator at Random.org, linked here, to generate a list of three fractions with four decimal places(LIST random decimal fraction generator  IS ON TOP). Assume those decimal fractions represent probability values associated with z-scores. Then use the standard normal table to look up the z-score that is closest to matching with that probability. List them below. (3 points)
   1. probability value (generated fraction) =
   2. associated z-score =
   3. probability value (generated fraction) =
   4. associated z-score =
   5. probability value (generated fraction) =
   6. associated z-score =
2. Use another random decimal fraction generator at Random.org, linked here, to generate a list of ten two-digit random numbers between 10 and 30. Calculate the z-score of the median of the data set. (3 points)
3. What does the z-score of the data set median just above tell you about the shape of the distribution? How do you know this? (3 points)
4. If you were to take repeated random samples of n = 5 from the data set just above, what would be the expected value of the mean of the sampling distribution of sample means? (3 points)
5. Considering the set of ten two-digit random numbers above as a population, calculate the standard error of the mean for the samples in question 4. (3 points)

Answer 1

Note - First four questions are solved.

Question 1
Use the random decimal fraction generator at Random.org, linked here, to generate a list of three fractions with four decimal places(LIST random decimal fraction generator IS ON TOP). Assume those decimal fractions represent probability values associated with z-scores. Then use the standard normal table to look up the z-score that is closest to matching with that probability. List them below. (3 points)

Steps to find zscore give the probability.
1. Find the value closest to the probability in the normal distribution table.
2. The zscore is obtained by adding the number on left with a number on the top as shown below.
3. The closed number to 0.1613 is 0.16109 hence the number of the left is -0.9 and the number on top is 0.09. We add the absolute value of the two 0.9 + 0.09 and add the negative sign since the left number carries it.

Hence the zscore of 0.1613 = -0.99

The zscore of the random number is given below.

Question 2
Use another random decimal fraction generator at Random.org, linked here, to generate a list of ten two-digit random numbers between 10 and 30. Calculate the z-score of the median of the data set. (3 points)

This problem is solved in the excel. The formulas used as also shown below.

Question 3
What does the z-score of the data set median just above tell you about the shape of the distribution? How do you know this? (3 points)

The zscore of the mean is equal to zero.
The zscore of the median is equal to -0.43.
Hence the median is less than the mean.
Therefore we can conclude the data is skewed to the right. Hence the shape of the distribution is right-skewed.

Question 4
If you were to take repeated random samples of n = 5 from the data set just above, what would be the expected value of the mean of the sampling distribution of sample means? (3 points)
From the Central limit theorem, we have known that the mean of the sampling means will be equal to the mean of the population as the number of samples increases.
Hence if we take repeated random samples of n = 5 from the dataset, the expected value of the mean of the sampling distribution of the sample mean, will be almost equal to the mean of the dataset.

Question 5

Considering the set of ten two-digit random numbers above as a population, calculate the standard error of the mean for the samples in question 4. (3 points)

Standard error =