Question

1. Answer the following critical thinking questions in this forum. Feel free to copy the questions and paste into your reply so that you cover each one. Use complete sentences and proper statistical terminology in all posts.
2. Read the instructors response to the questions.
3. Read the answers of some of your peers.

Critical Thinking Questions

1. What does a z-score tell you about a number in a data set? [1 sentence]
2. What two quantities do we need to fully describe a normal distribution? [1 sentence]
3. How is probability determined from a continuous distribution? Why is this easy for the uniform distribution and not so easy for the normal distribution? [2 sentences]
4. What does the symmetric bell shape of the normal curve imply about the distribution of individuals in a normal population? [2 sentences]
5. How can the empirical rule be restated in terms of z-scores and percentiles? Restate it for four of the seven z-scores. Hint: Use the definitions of z-score and percentile and avoid use of the phrase "standard deviation" or the numbers 68, 95, and 99.7. [4 statements]

Answer 1

What does a z-score tell you about a number in a data set? [1 sentence]

The z-score gives the relative position of the data point on the standard normal distribution curve from the mean such that how many standard deviations a data point lies on the standard normal distribution curve from the mean.

What two quantities do we need to fully describe a normal distribution? [1 sentence]

A normal distribution curve is described by two-parameter i.e. mean and the standard deviation.

How is probability determined from a continuous distribution? Why is this easy for the uniform distribution and not so easy for the normal distribution? [2 sentences]

The probability of a continuous distribution is obtained by integrating the area which falls under the specified limit.

Now, calculating the probability for the uniform distribution is easy because taking the integration of uniform distribution is simple compared to the Gaussian function for normal distribution.

What does the symmetric bell shape of the normal curve imply about the distribution of individuals in a normal population? [2 sentences]

If the distribution for the individual data point is symmetric we can say that the shape of the distribution is the same on either side of the mean.

For the symmetric distribution, the mean, median and mode all lie at the same point i.e. in the middle of the distribution.

How can the empirical rule be restated in terms of z-scores and percentiles? Restate it for four of the seven z-scores.

The z score tells how far a data point lies from the mean of the normal curve. It uses mean as the center of the distribution. The percentile score of the data point tells the percentage of the data point that falls under it.

The empirical rule says that if the distribution is roughly normal then approximately 68%, 95% and 99.7% data point lies within one, two and three standard deviations from mean respectively.

We can restate the empirical rule in terms of z scores as

Within z = 2

for z score = -2, percentile score = 2.28% such that 2.28% data value lies below z score = -2 and for z score = 2, percentile score = 97.72% such 97.72% data value lies below z score = 2 and within z = 2 approximately 95% data value lies

Within z = 1

for z score = -1, percentile score = 15.87% such that 15.87% data value lies below z score = -1 and for z score = 1, percentile score = 87.13% such 84.13% data value lies below z score = 1 and within z = 1 approximately 68% data value lies.