Z score statcrunch

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Chapter 5: Normal Probability Distributions

Section 5.1: Intro to Normal Distributions and the Standard Normal Distributions Objectives:

 Normal Distribution Properties

 Use z-scores to Calculate Area Under the Standard Normal Curve (using StatCrunch or Calculator)

 Discuss Unusual Values

In this section we will revisit histograms which can be estimated with normal (symmetric, bell-shaped) curves. From Test 1 remember that normal curves have z-scores (for any data value) and areas under the curve (one way: Empirical Rule). Now we will use these normal curves to find probabilities (areas) and z-scores for any data value. Why do we need to study this? Eventually we will use these probabilities and z-scores to make decisions. By using the normal distribution curve, we are treating the data as a continuous random variable that has its own continuous probability distribution. (Remember that any probability distribution has two properties: all probabilities are between 0 and 1 and the sum of the probabilities is 1.) **Probabilities = Areas under the curve**

Ex: Consider the normal distribution curves below. Which normal curve has the greatest mean? Which normal curve has the greatest standard deviation?

Note: Every normal distribution can be transformed into the Standard Normal Distribution (the distribution for z- scores). This means we will use the z-score formula to transform any data value into a “measure of position” with the formula:

data value mean

standard deviation z

 

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**All probability calculations will be done with either StatCrunch or the TI 83/84 calculator. You do NOT need to learn how to read the Standard Normal Table.**

**Also < and  are treated the same as well as > and  for any continuous probability distribution.**

Ex: Confirm that the area to the left of z = 1.15 is 0.8749. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -1000000000  Comma  1.15 Comma  0  Comma 1 enter P(z  1.15) = 0.8749 Ex: Confirm that the cumulative area that corresponds to z = -0.24 is 0.4052. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -1000000000  Comma  -0.24 Comma 0  Comma 1 enter P(z  -0.24) = 0.4052

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Ex: Find the area to right of each z-score. Hint: Use the fact that the total area (probability) is 1. **Label the z-score and the area.** a) b)

P(z  1.15) = _________________ P(z  -0.24) = _________________ Ex: Find the shaded area. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -2.3  Comma  1000000000 Comma 0  Comma 1 enter

P(z  -2.3) = _________________ Ex: Find the shaded area. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Between, enter z-scores, Compute TI-83/84: 2nd  VARS normalcdf( -1.5  Comma  1.25 Comma 0  Comma 1 enter

P(-1.5  z  1.25) = _________________ Ex: The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed. In a recent year, the mean ( ) test score was 1498 and the standard deviation ( ) was 316. The

test scores of four students selected at random are 1920, 1240, 2350, and 1390. Sketch this distribution, find the z-scores for each value, and determine whether any of the values are unusual when compared to the mean and standard deviation.

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Section 5.2: Normal Probability Distributions: Finding Probabilities Objectives:

 Sketch Normal Distribution along with Standard Normal Distribution

 Find z-scores to calculate Area Under the Normal Curve (using StatCrunch or Calculator)

In this section you will get the chance to apply the probabilities (areas) from the Standard Normal Distribution to real- life situations. Consider the last example concerning the SAT: Ex: The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed. In a recent year, the mean ( ) test score was 1498 and the standard deviation ( ) was 316. The

test scores of four students selected at random are 1920, 1240, 2350, and 1390. Sketch this distribution and find the area to the right (probability above) for each score. Are any of these events unusual? **Remember: Any probability that is 0.05 or below means that an event is considered unusual… Student 1: score = 1920 Student 2: score = 1240 z-score = _______________ z-score = _______________ P(x > 1920) = P(z > _____ ) = ________________ P(x > 1240) = P(z > _____ ) = ________________ Unusual Event? ________________ Unusual Event? ________________ Student 3: score = 2350 Student 4: score = 1390 z-score = _______________ z-score = _______________ P(x > 2350) = P(z > _____ ) = ________________ P(x > 1390) = P(z > _____ ) = ________________ Unusual Event? ________________ Unusual Event? ________________ StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & standard deviation, enter inequality symbol and x value, Compute TI-83/84: 2nd  VARS normalcdf( x value  Comma  1000000000 Comma mean  Comma standard deviation enter