# Z-Scores, the Central Limit Theorem and areas under a Normal Curve

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This paper answers questions about calculating z-scores from raw data values as well as figuring out what the raw data values must have been if you know what the z-scores are. It also deals with calculation of the standard error of a mean for various sample sizes and calculating certain areas under a normal, bell-shaped curve.

Statistics: Normal curve, normal distribution, probability, standard deviation, z-score. Finding area under a curve between two z-scores.

Please find below some problems to be solved.

1. Find the area of the indicated region under the standard normal curve. The area between z=0 and z=1.1 under the standard normal curve is _________. (Round to four decimal places as needed.)

2. Find the area of the indicated region under the standard normal curve. The area between z= -1 and z=0.8 under the standard normal curve is _________ (Round to four decimal places as needed).

3. Find the indicated area under the standard normal curve is ________. To the right of z= 1.01.

The area to the right of z = 1.01 under the standard normal curve is ________. (Round to four decimal places as needed.)

Between z = 0 and z= 2.28

The area between z=0 and z= 2.28 under the standard normal curve is ______. (Round to four decimal places as needed (Round to four decimal places as needed.)

Find the indicated area to the left of z=-2.58 or to the right of z= 2.58 under the standard normal curve is ____________.

For the standard normal distribution shown on the right, find the probability of z occurring in the indicated region. (Indicated region to the left is -2.09).

The probability is __________. (Round to four decimal places as needed).

P (-0.87 less than z less than 0.87) = ___________ (Round to four decimal places as needed).

Find the indicated probability using the standard normal distribution.

P (-0.87 less than z less than 0.87).

P (-0.87 less than z less than 0.87 =_________ (Round to four decimal places as needed).

The random -number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution.

What is the probability of generating a number between 0.23 and 0.75?

What is the probability of generating a number density function below?

Density

1.2

1-

0.8-

0.6-

0.4-

0.2-

Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable x is normally distributed.

Sat critical Reading Scores

200 less than x less than 450

Left side shaded 200

Score: 450

X side 800

The Mean= 511

Standard deviation= 111

The probability that the member selected at the shaded area of the graph is ______ (round to four decimal places as needed.

In recent year, scores on a standardized test for high school students with a 3.50 to 4.00 grade point average were normally distributed, with a mean of 37.8 and a standard deviation of 2.5. A student with a 3.50 to 4.00 grade point average who took the standardized test is randomly selected.

(a) Find the probability that the student's test score is less than 34.

The probability of a student scoring less than 34 is _______. (Round to four decimal places).

(b) Find the probability of a student's test score between 34.5 and 41.1.

The probability of a student scoring between 34.5 and 41.1 is _________ (Round to four decimal places as needed.)

(c) Find the probability that the student's test score is more than 39.5.

The probability of a student scoring more than 39.5 is ________. (Round to four decimal places as needed).

The life span of a battery is normally distributed with a mean of 2200 hours and a t percent of batteries have a life standard deviation of 60 hours. What percent of batteries have a life span that is more than 2275 hours? Would it be unusual for a battery to have a life span that is more than 2275 hours? Explain your reasoning.

What percent of batteries have a life span more than 2275 hours?

Approximately _____% of batteries have a life span that is more than 2275 hours. (Round to two decimal places as needed).

Would it be unusual for a battery to have a life span that is more than 2275 hours? Explain your reasoning.

A. It is not unusual for a battery to have a life span that is more than 2275 hours because the z-score is not within 2 standard deviations of the mean.

B. It is not unusual for a battery to have a life span that is more than 2275 hours because the z-score is within 2 standard deviations of the mean.

C. It is unusual for a battery to have a life span that is more than 2275 hours because the z-score is not within 2 standard deviations of the mean.

D. It is not unusual for a battery to have a life span that is more than 2275 hours because the z-score is within 2 standard deviations of the mean

Use the standard normal table to find the z-score that corresponds to the given percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use the technology to find the z-score.

P 2

The z-score that corresponds to P square is ______ Round to three decimal places.

Find the indicated z-score on a graph which has an area = 0.2643 0n the left of a graph.

Z= ?

The z -score is __________ (Round to two decimal places).

Find the indicated z-scores which are shown in a graph which has an area of 0.0307 0n the left of the graph and 0.0307 shown on the right of the graph.

The z-score is ______. (Use a comma to separate answers as needed. Round to two decimal places as needed).

In a survey of women in a certain country (ages 20-29), the mean height was 65.7 inches with a standard deviation of 2.91 inches. Answer the following questions about the specified normal distribution.

(a) What height represents the 85th percentile?

(b) What height represents the first quartile?

(a) The height that represents the 85th percentile is ______inches (Round to two decimal places as needed).

(b) The height that represents the first quartile is ______inches. (Round to two decimal places as needed.)

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.

For a sample of n=75, find the probability of a sample mean greater than 211 if the mean = 210 and the standard deviation is =3.8.

For a sample of n=75, the probability of a sample mean being greater than 211 if mean= 210 and standard deviation = 3.8 is _________. (Round to four decimal places as needed.)

Would the given sample mean be considered unusual?

The sample mean ________be considered unusual because it _______ within ________of the mean of the sample means.

The height of fully grown trees of a specific species is normally distributed, with a mean of 71.0 feet and a standard deviation of 7.25 feet. Random samples of size 18 are drawn from the population. Use the central limit theorem to find the mean and standard error of the sampling distribution. Then sketch a graph of the sampling distribution.

The mean of the sampling distribution is ________.

The standard error of the sampling distribution is standard deviation x -= _______. (Round to two decimal places as needed).

Determine if the finite correction factor should be used. If so, use it in your calculations when you find the probability.

In a sample of 700 gas stations, the mean price for regular gasoline at the pump was $2.812 per gallon and the standard deviation was $0.009 per gallon. A random sample of size 60 is drawn from this population. What is the probability that the mean price per gallon is less than $2.811/

The probability that the mean price per gallon is less than $2.811 is ______. (Round to four decimal places as needed).

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