MAT 120 Section 5.1 Introduction to Normal Distribution

Find the indicated probability using the standard normal distribution. ​P(z < −1.65 or z > 1.65​)

ANSWER: Work: On the Chart (Area under the standard normal distribution to the left of z) z = -1.65 is 0.0495 z = 1.65 is 0.9505 1 - 0.9505 = 0.0495 0.0495 + 0.0495 = 0.0990.

Complete Parts a and b. a) What do the inflection points on a normal distribution​ represent? Choose the correct answer below. A. They are the points that mark the boundaries of the middle​ 50% of the area under the curve. B. They are the points at which the curve changes between curving upward and curving downward. C. They are the points at which the curve changes sign. D. They are the points at which the cumulative area under the curve is 0 and 1. b) Where do they​ occur? Choose the correct answer below. A. μ - σ and μ + σ B. μ - 2σ and μ + 2σ C. μ - 3σ and μ + 3σ D. μ - σ, μ + σ, μ - 3σ, and μ + 3σ

ANSWER: a) B. They are the points at which the curve changes between curving upward and curving downward. b) A. μ - σ and μ + σ. Explanation for a)The normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. Between the inflection points of the​ curve, the graph curves downward. Outside of the inflection​ points, the graph curves upward. Explanation for b) The normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. Between the points μ−σ and μ+σ​, the graph curves downward. Outside of the inflection​ points, the graph curves upward. The points at which the curve changes between curving upward and curving downward are called inflection points.

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences. Compare the two curves. The two curves will have ____?the same line/different lines? ____ of symmetry. The curve with the larger standard deviation will be ____?less/more?____ spread out than the curve with the smaller standard deviation.

ANSWER: -The same line. -More. The two curves will have the same line of symmetry because the means are equal. The curve with the larger standard deviation will be more spread out than the curve with the smaller standard deviation because a larger standard deviation is a result of an increased distance from the mean.

Find the indicated probability using the standard normal distribution. ​P(z < −2.71​)

ANSWER: 0.0034 Work: on the chart (Area under the standard normal distribution to the left of z) z = -2.71 is 0.0034.

Find the indicated probability using the standard normal distribution. ​P(z > 2.54​)

ANSWER: 0.0055. Work On the Chart (Area under the standard normal distribution to the left of z) z = 2.54 is 0.9945 1 - 0.9945 = 0.0055

Find the indicated area under the standard normal curve. Between z=0 and z=2.17 The area between z=0 and z=2.17 under the standard normal curve is ___?

ANSWER: 0.4850. work: on the chart (Area under the standard normal distribution to the left of z) z=0 is 0.5000 on chart. z=2.17 is 0.9850 on chart 0.9850 - 0.5000 = 0.4850.

Find the indicated probability using the standard normal distribution. ​P(−2.83 < z < 0​)

ANSWER: 0.4977. Work: On the Chart (Area under the standard normal distribution to the left of z) z = -2.83 is 0.0023 z = 0 is 0.5000. 0.5000 - 0.0023 = 0.4977

Find the indicated area under the standard normal curve. To the left of z=−0.38 and to the right of z=0.38 The total area to the left of z=−0.38 and to the right of z=0.38 under the standard normal curve is ___?

ANSWER: 0.7040 Work: On the Chart (Area under the standard normal distribution to the left of z) z= -0.38 is 0.3520 z= 0.38 is 0.6480 1 - 0.6480 = 0.3520 0.3520 + 0.3520 = 0.7040.

Find the indicated area under the standard normal curve. Between z=−1.81 and z=1.81 The area between z=−1.81 and z=1.81 under the standard normal curve is ____?

ANSWER: 0.9298. Work on chart (Area under the standard normal distribution to the left of z) z=−1.81 is 0.0351 z=1.81 is 0.9649 0.9649 - 0.0351 = 0.9298.

For the standard normal distribution shown on the​ right, find the probability of z occurring in the indicated region. z = -1.69

ANSWER: 0.9545 Work on the chart (Area under the standard normal distribution to the left of z) z = -1.69 is 0.0455 1 - 0.0455 = 0.9545

Find the indicated probability using the standard normal distribution. ​P(−2.79 < z < 2.79​)

ANSWER: 0.9948. Work: On the Chart (Area under the standard normal distribution to the left of z) z = -2.79 is 0.0026 z = 2.79 is 0.9974 0.9974 - 0.0026 = 0.9948

In a normal​ distribution, which is​ greater, the mean or the​ median? Explain. Choose the correct answer below. A. The​ mean; in a normal​ distribution, the mean is always greater than the median. B. Neither; in a normal​ distribution, the mean and median are equal. C. The​ median; in a normal​ distribution, the median is always greater than the mean.

ANSWER: B. Neither; in a normal​ distribution, the mean and median are equal. One of the properties of a normal distribution is that the​ mean, median, and mode are equal.

What requirements are necessary for a normal probability distribution to be a standard normal probability​ distribution? Choose the correct answer below. A. The mean and standard deviation have the values of μ=1 and σ=1. B. The mean and standard deviation have the values of μ=0 and σ=0. C. The mean and standard deviation have the values of μ=0 and σ=1. D. The mean and standard deviation have the values of μ=1 and σ=0.

ANSWER: C. The mean and standard deviation have the values of μ=0 and σ=1.

What is the total area under the normal​ curve? Choose the correct answer below. A. 0.5 B. It depends on the mean. C. It depends on the standard deviation. D. 1

ANSWER: D. 1.

Describe how you can transform a nonstandard normal distribution to the standard normal distribution. To transform a nonstandard normal distribution to the standard normal distribution you must transform each data value x into a​ z-score. Which of the following formulas is used to convert an x value into a​ z-score? A. z = (x-μ)^2/σ B. z = x-μ C. z = (μ-x)/σ^2 D. z = (x-μ)/σ

ANSWER: D. z = (x-μ)/σ. Transform each data value x into a​ z-score by subtracting the mean from x and dividing the result by the standard deviation.

For the standard normal distribution shown on the​ right, find the probability of z occurring in the indicated region. z = 0.11.

ANSWER: 0.5438 work on chart (Area under the standard normal distribution to the left of z) z=0.11 is 0.5438.