NAME:______BLOCK:______DATE:______
6.1 Graphs of Normal Probability Distributions
Important Properties of a Normal Curve:
1. The curve is bell-shaped with the highest point over the mean µ
2. It is symmetric about the vertical line through µ
3. The curve approaches the horizontal axis but never crosses or touches it.
4. The transition points are where the graph changes from cupping upward to cupping downward (or visa versa). The transition points occur at x-μ and x+μ
5. The total area under the curve is 1.
Exercise 1: Which normal curve has the greater mean?Exercise 2: Which normal curve has the greater standard deviation?
Exercise 3: The playing life of a Sunshine radio is normally distributed with a mean of 600 hours and a standard deviation of 100 hours. Sketch a normal curve showing the distribution of the playing life of the Sunshine radio. Scale and label the axis; include the transition points.
Exercise 4: The playing life of a Sunshine radio is normally distributed with a mean of 600 hours and a standard deviation of 100 hours. Use the empirical rule to compute the probabilities that a randomly selected radio will last as specified:
Life of Randomly Selected Radio / Probability Expression / Probability CalculationBetween 600 and 700 hours / P(600 ≤x ≤700)
Between 400 and 500 hours
Greater than 700 hours
Exercise 5: The annual wheat yield per acre on a farm is normally distributed with a mean of 35 bushels and a standard deviation of 8 bushels. Sketch a normal curve and shade in the area that represents the probability that an acre will yield between 19 and 35 bushels. Use the empirical rule to find the probability that the yield will be between 19 and 35 bushels per acre.
6.2 Areas Under the Standard Normal Distribution
z- Score Conversion Formulas
Convert raw score to a score
/z= x- μσ
Convert z-score to raw score
/x=zσ+ μ
Exercise 1 Use the standard normal table to compute each of the cumulative areas below.
/Description
/Cumulative Area
a) / Find the cumulative area that corresponds to a z-score of 1.15. /b) / Find the cumulative area that corresponds to a z-score of -0.24. /
c) / Find the cumulative area that corresponds to a z-score of -0.99. /
d) / Find the cumulative area to the right of z=1.06 /
e) / Find the area under the standard normal curve between z = –1.5 and z = 1.25. /
Exercise 2 Re-compute Exercise 1 (parts a-e) using the normalcdf function on the TI-84 calculator.
Exercise 3 Suppose Tina and Jack are in two different sections of the same course and they recently took midterms. Tina’s class average was 64 (with S.D.=3) and she got a 74. Jack’s class average was 72 (with S.D.=5) and he got an 82. Assuming that all scores are normally distributed, who did better relative to the class?
Interpretive Statement:
Exercise 4 A pizza parlor chain claims a large pizza has 8 oz. of cheese with a standard deviation of 0.5 oz. An inspector ordered a pizza and found it only had 6.9 oz. of cheese. Franchisee’s can lose their store if they make pizzas with 3 standard deviations (or more) of cheese below the mean. Assume the distribution of weights is normally distributed.
a) Find the z-score for x = 6.9 oz. of cheese.
b) Is the franchisee in danger of losing its store? Why?
c) Find the minimum amount of cheese a franchise can put on a large pizza so it is not in danger of losing its store.
6.3 Areas Under Any Normal Curve
Exercise 1 A survey indicates that people use their cellular phones an average of 1.5 years before buying a new one. The standard deviation is 0.25 year. A cellular phone user is selected at random. Find the probability that the user will use their current phone for less than 1 year before buying a new one. Assume that the variable x is normally distributed.
Exercise 2 A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. Find the probability that the shopper will be in the store for between 24 and 54 minutes.
Exercise 3 If 200 shoppers enter the store, how many shoppers would you expect to be in the store between 24 and 54 minutes?
Exercise 4 Find the probability that the shopper will be in the store more than 39 minutes. (Recall μ = 45 minutes and σ = 12 minutes)
Exercise 5 If 200 shoppers enter the store, how many shoppers would you expect to be in the store more than 39 minutes?
Exercise 6 Repeat Exercise 5 using the TI-84 calculator.
6.3 Inverse Normal Lookups
Exercise 1 Find the z-score that corresponds to a cumulative area of 0.3632. Solve using the Standard Normal Table and the TI-84 calculator.
Exercise 2 Find the z-score that has 10.75% of the distribution’s area to its right.
Exercise 3 Find the z-score that corresponds to a cumulative area of 0.3632.
Exercise 4 Find the z-score that corresponds to P5
Exercise 5 Find the z value such that 90% of the area under the standard normal curve lies between –z and z.
Exercise 6 A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z-scores of 1.96, –0.44, and 0.
Exercise 7 Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency?
Exercise 8 Magic Video Games Inc. sells expensive computer games and wants to advertise an impressive, full-refund warranty period. It has found that the mean life for its computer games is 30 months with a standard deviation of 4 months. If the life spans of the computer games are normally distributed, how long of a warranty period (to the nearest month) can be offered so that the company will not have to refund the price of more than 7% of the computer games?
Interpretive Statement:
6.4 Normal Approximations to Binomial Distributions
Exercise 1 Use a continuity correction to convert the binomial interval to a normal distribution interval: The probability of getting between 270 and 310 successes, inclusive.
Exercise 2 Use a continuity correction to convert the binomial interval to a normal distribution interval: The probability of getting at least 158 successes.
Exercise 3 Use a continuity correction to convert the binomial interval to a normal distribution interval: The probability of getting fewer than 63 successes.
Exercise 4 Sixty-two percent of adults in the U.S. have an HDTV in their home. You randomly select 45 adults in the U.S. and ask them if they have an HDTV in their home. What is the probability that fewer than 20 of them respond yes?
Exercise 5 A survey reports that 62% of Internet users use Google Chrome as their browser. You randomly select 150 Internet users and ask them whether they use Chrome as their browser. What is the probability that exactly 96 will say yes?
Exercise 6 The owner of a new apartment building needs to have 25 new water heaters installed. Assume the probability that a water heater will last 10 years is 0.25.
a) What is the probability that 8 or more will last at least 10 years? Use the binomial distribution.
b) Can the binomial probability distribution be approximated by a normal distribution? Explain.
c) If so, use a normal distribution to approximate the binomial distribution
c) Find the error between the two calculations.
Practice #1
Practice #2
Practice #1: / Find the area under the standard normal curveLower / Upper / Area Type
Bound / Bound / Left Tail / Right Tail / Between
1) / -3 / 3 / 0.9973
2) / 1 / 0.1587
3) / 0 / 2.53 / 0.4943
4) / 2.53 / 0.0057
5) / -2.34 / 0.0096
6) / -2 / 2 / 0.9545
Practice #2: / Inverse Lookups
Find the specified Value
Area / Left Tail / Right Tail / Center Area
µ / σ / Components / Area / Area / A / (1-A)/2 / variable / value / value
a) / 0.0 / 1.0 / 0.50 / 0.32 / 0.82 / a = / 0.92
b) / 0.0 / 1.0 / 0.94 / 0.03 / a = / -1.88
c) / 90.0 / 7.0 / 0.82 / 0.09 / z = / -1.34 / 1.34
90.0 / 7.0 / bL / 80.61
90.0 / 7.0 / bR / 99.39
d) / 45.0 / 5.0 / 0.88 / 0.06 / z = / -1.55 / 1.55
45.0 / 5.0 / xL / 37.23
45.0 / 5.0 / xR / 52.77
45.0 / 5.0 / b = / 7.77
12