Chapter 5 - Normal Probabilities and the Standard Normal Distribution

Given the mean of a normal​ distribution, how can you find the​ median?

When it is a normal​ distribution, the mean is equal to the median.

What is the total area under the normal​ curve?

1

Describe the inflection points on the graph of a normal distribution. At what​ x-values are the inflection points​ located?

They are the points at which the curve changes between curving upward and curving downward.

Part 2

Mean + Standard Deviation

Mean - Standard Deviation

Give two​ real-life examples of a continuous variable. Include one that is likely to be normally distributed. Explain your reasoning.

Temperature (C Degrees°​) in a school is a continuous​ variable, because it has infinite possible outcomes.

​Student's height in a school is likely to be a normally distributed and continuous​ variable, because theoretically height can be measured with infinitly increasing​ accuracy, such as​ 61", 61.03", or​ 61.028800001", and the likelihoods of the possible results form a bell curve.

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

The two curves will have the same line of symmetry. The curve with the larger standard deviation will be more spread out than the curve with the smaller standard deviation.

What is the mean of the standard normal​ distribution? What is the standard deviation of the standard normal​ distribution?

The mean and standard deviation have the values of mean = 0 and standard deviation = 1

Why is it correct to say​ "a" normal distribution and​ "the" standard normal​ distribution?

​"The" standard normal distribution is used to describe one specific normal distribution

left parenthesis mu equals 0 comma sigma equals 1 right parenthesis .(μ=0, σ=1).

​"A" normal distribution is used to describe a normal distribution with any mean and standard deviation.

If a​ z-score is​ zero, which of the following must be​ true? Explain your reasoning.

The corresponding​ x-value is equal to the​ mean, because the​ z-score is equal to the difference between the​ x-value and the​ mean, divided by the standard deviation.

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

The two curves will have same of symmetry. The curve with the larger standard deviation will be more spread out than the curve with the smaller standard deviation.

As the size of a sample​ increases, the mean of the distribution of sample means increases.

False. As the size of a sample​ increases, the mean of the distribution of sample means does not change.

As the size of a sample​ increases, the standard deviation of the distribution of sample means increases.

This statement is false. A true statement​ is, "As the size of a sample​ increases, the standard deviation of the distribution of sample means​ decreases."

A sampling distribution of a sample mean is normal only if the population is normal.

The statement is false. Sampling distribution of a sample mean is approximately normal if either greater than or equals n ≥30 or the population is normal.

If the size of a sample is at least​ 30, then you can use​ z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.

The statement is true