The shaded area in the following graph indicates the area to the left of (x)" /> The shaded area in the following graph indicates the area to the left of (x)" />

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The shaded area in the following graph indicates the area lớn the left of (x). This area is represented by the probability (P(X

Figure (PageIndex1).

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The area khổng lồ the right is then (P(X > x) = 1 – P(X x) = 1 – P(X x)) is the same as (P(X geq x)) for continuous distributions.

Example (PageIndex1)

If the area khổng lồ the left is 0.0228, then the area to the right is (1 - 0.0228 = 0.9772).

Exercise (PageIndex1)

If the area to lớn the left of (x) is (0.012), then what is the area khổng lồ the right?

(1 - 0.012 = 0.988)

Example (PageIndex2)

The final exam scores in a statistics class were normally distributed with a mean of 63 và a standard deviation of five.

Find the probability that a randomly selected student scored more than 65 on the exam. Find the probability that a randomly selected student scored less than 85. Find the 90th percentile (that is, find the score (k) that has 90% of the scores below k và 10% of the scores above (k)). Find the 70th percentile (that is, find the score (k) such that 70% of scores are below (k) & 30% of the scores are above sầu (k)).

a. Let (X) = a score on the final exam. (X sim N(63, 5)), where (mu = 63) & (sigma = 5)

Draw a graph.

Then, find (P(x > 65)).

65) = 0.3446 onumber >

Figure (PageIndex2).

The probability that any student selected at random scores more than 65 is 0.3446.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

Go into 2nd DISTR.

After pressing 2nd DISTR, press 2:normalcdf.

The syntax for the instructions are as follows:

normalcdf(lower value, upper value, mean, standard deviation) For this problem: normalcdf(65,1E99,63,5) = 0.3446. You get 1E99 (= 1099) by pressing 1, the EE key (a 2nd key) and then 99. Or, you can enter 10^99instead. The number 1099 is way out in the right tail of the normal curve sầu. We are calculating the area between 65 và 1099. In some instances, the lower number of the area might be –1E99 (= –1099). The number –1099 is way out in the left tail of the normal curve sầu.

Historical Note

The TI probability program calculates a (z)-score và then the probability from the (z)-score. Before công nghệ, the (z)-score was looked up in a standard normal probability table (because the math involved is too cumbersome) lớn find the probability. In this example, a standard normal table with area to lớn the left of the (z)-score was used. You calculate the (z)-score và look up the area lớn the left. The probability is the area to the right.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

Find the percentile for a student scoring 65:

*Press 2nd Distr*Press 2:normalcdf(*Enter lower bound, upper bound, mean, standard deviation followed by )*Press ENTER.For this Example, the steps are2nd Distr2:normalcdf(65,1,2nd EE,99,63,5) ENTERThe probability that a selected student scored more than 65 is 0.3446.To find the probability that a selected student scored more than 65, subtract the percentile from 1.

b. Draw a graph.

Then find (P(x th percentile. For each problem or part of a problem, draw a new graph. Draw the (x)-axis. Shade the area that corresponds lớn the 90th percentile.

Let (k =) the 90th percentile. The variable (k) is located on the (x)-axis. (P(x th percentile (k) separates the exam scores inkhổng lồ those that are the same or lower than (k) và those that are the same or higher. Ninety percent of the demo scores are the same or lower than (k), và ten percent are the same or higher. The variable (k) is often called a critical value.

(k = 69.4)

Figure (PageIndex3).

The 90th percentile is 69.4. This means that 90% of the test scores fall at or below 69.4 và 10% fall at or above sầu. To get this answer on the calculator, follow this step:

invNorm in 2nd DISTR. invNorm(area to the left, mean, standard deviation)

For this problem, ( extinvNorm(0.90,63,5) = 69.4)

d. Find the 70th percentile.

Draw a new graph and label it appropriately. (k = 65.6)

The 70th percentile is 65.6. This means that 70% of the test scores fall at or below 65.5 and 30% fall at or above.

( extinvNorm(0.70,63,5) = 65.6)

Exercise (PageIndex2)

The golf scores for a school team were normally distributed with a mean of 68 và a standard deviation of three. Find the probability that a randomly selected golfer scored less than 65.

Xem thêm: Tiếng Anh Lớp 5 Tập 2 Unit 14, Tiếng Anh 5 Unit 14 Lesson 2 (Trang 26

( extnormalcdf(10^99,65,68,3) = 0.1587)

Example (PageIndex3)

A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed & the standard deviation for the times is half an hour.

Find the probability that a household personal computer is used for entertainment between 1.8 và 2.75 hours per day. Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment.

a. Let (X =) the amount of time (in hours) a household personal computer is used for entertainment. (X syên ổn N(2, 0.5)) where (mu = 2) and (sigma = 0.5).

Find (P(1.8

Figure (PageIndex4).

< extnormalcdf(1.8,2.75,2,0.5) = 0.5886 onumber >

The probability that a household personal computer is used between 1.8 và 2.75 hours per day for entertainment is 0.5886.

b.

To find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment, find the 25th percentile, (k), where (P(x

Figure (PageIndex5).

< extinvNorm(0.25,2,0.5) = 1.66 onumber >

The maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment is 1.66 hours.

Exercise (PageIndex3)

The golf scores for a school team were normally distributed with a mean of 68 và a standard deviation of three. Find the probability that a golfer scored between 66 & 70.

( extnormalcdf(66,70,68,3) = 0.4950)

Example (PageIndex4)

There are approximately one billion điện thoại thông minh users in the world today. In the United States the ages 13 to lớn 55+ of smartphone users approximately follow a normal distribution with approximate mean & standard deviation of 36.9 years and 13.9 years, respectively.

Determine the probability that a random smartphone user in the age range 13 to 55+ is between 23 & 64.7 years old. Determine the probability that a randomly selected smartphone user in the age range 13 lớn 55+ is at most 50.8 years old. Find the 80th percentile of this distribution, và interpret it in a complete sentence.

( extnormalcdf(23,64.7,36.9,13.9) = 0.8186) ( extnormalcdf(-10^99,50.8,36.9,13.9) = 0.8413) ( extinvNorm(0.80,36.9,13.9) = 48.6)

The 80th percentile is 48.6 years.

80% of the smartphone users in the age range 13 – 55+ are 48.6 years old or less.

Exercise (PageIndex4)

Find the 30th percentile, & interpret it in a complete sentence. What is the probability that the age of a randomly selected điện thoại thông minh user in the range 13 to lớn 55+ is less than 27 years old.

70.

Let (X =) a smart phone user whose age is 13 to lớn 55+. (X syên ổn N(36.9, 13.9))

To find the 30th percentile, find (k) such that (P(x ( extinvNorm(0.30, 36.9, 13.9) = 29.6) years Thirty percent of smartphone users 13 lớn 55+ are at most 29.6 years & 70% are at least 29.6 years. Find (P(x (Note that ( extnormalcdf(-10^99,27,36.9,13.9) = 0.2382). The two answers differ only by 0.0040.)

Exercise (PageIndex6)

Using the information from Example, answer the following:

The middle 45% of mandarin oranges from this farm are between ______ & ______. Find the 16th percentile và interpret it in a complete sentence. Answer a

The middle area (= 0.40), so each tail has an area of 0.30.

( – 0.40 = 0.60)

The tails of the graph of the normal distribution each have an area of 0.30.

Find (k1), the 30th percentile and (k2), the 70th percentile ((0.40 + 0.30 = 0.70)).

(k1 = extinvNorm(0.30,5.85,0.24) = 5.72) cm

(k2 = extinvNorm(0.70,5.85,0.24) = 5.98) cm

( extnormalcdf(5,10^99,5.85,0.24) = 0.9998)

Exercise (PageIndex7)

How would you represent the area to the left of one in a probability statement?

Figure (PageIndex8).

(P(x Exercise (PageIndex8)

Is (P(x Exercise (PageIndex9)

How would you represent the area khổng lồ the left of three in a probability statement?

Figure (PageIndex10).

Xem thêm: Lời Bài Hát Nói Chia Tay Thật Khó Nói, Lời Bài Hát Lời Chia Tay Khó Nói (Vương Khang)

Exercise (PageIndex10)

What is the area lớn the right of three?

Exercise (PageIndex21)

Find the probability that a CD player will last between 2.8 và six years.

Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.

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