Psychology A test devised to measure aggressive–

passive personalities was standardized on a large group of people. The scores were normally distrib- uted, with a mean of 50 and a standard deviation of 10. If we want to designate the highest 10% as aggressive, the next 20% as moderately aggressive, the middle 40% as average, the next 20% as moderately passive, and the lowest 10% as passive, what ranges of scores will be covered by these five designations?

Chapter 8 Review

Important Terms, Symbols, and Concepts 8-1 Graphing Data

Bar graphs, broken-line graphs, and pie graphs are used to

present visual interpretations or comparisons of data. Large sets of quantitative data can be organized in a frequency

table, generally constructed by choosing five to twenty class intervals of equal length to cover the data range. The num-

ber of measurements that fall in a given class interval is called the class frequency, and the set of all such frequen- cies associated with their respective classes is called a

frequency distribution. The relative frequency of a class is

its frequency divided by the total number of items in the data set.

A histogram is a vertical bar graph used to represent a frequency distribution. A frequency polygon is a broken- line graph obtained by joining successive midpoints of the

tops of the bars in a histogram. A cumulative frequency

polygon, or ogive, is obtained by plotting the cumulative fre-

quency over the upper boundary of the corresponding class.

8-2 Measures of Central Tendency

• Mean

Ungrouped Data: If is a set of n measure- ments, then the mean is given by

The mean is denoted by if the data set is a sample, and by if the data set is an entire population.

Grouped Data: If a data set of n measurements is grouped into k classes, and is the midpoint of the ith class inter- val and is the ith class frequency, then the mean for thefi

xi  x [mean] = x1 + x₂ + p + x_n n x1, x2, p , xn

Chapter 8 Review 549

grouped data is given by

The mean for grouped data is denoted by if the data set is a sample, and by if the data set is an entire population.

• Median

Ungrouped Data: Arrange the measurements in ascend- ing or descending order. If the number of measurements is odd, the median is the middle measurement. If the number of measurements is even, the median is the mean of the two middle measurements.

Grouped Data: The median for grouped data with no classes of frequency zero is the number such that the histogram has the same area to the left of the median as to the right of the median.

• Mode The mode is the most frequently occurring mea-

surement in a data set. There may be a unique mode, several modes, or, if no measurement occurs more than once, essentially no mode.

8-3 Measures of Dispersion

• Range The range for a set of ungrouped data is the dif-

ference between the largest and the smallest values in the data set. The range for a frequency distribution is the dif- ference between the upper boundary of the highest class and the lower boundary of the lowest class.

• Standard Deviation

Ungrouped Data: The sample standard deviation s of a set of n sample measurements with mean is given by

The square of the sample standard deviation, is called the sample variance.

Grouped Data: Suppose a data set of n sample measure- ments is grouped into k classes in a frequency table, where

is the midpoint, is the frequency of the ith class interval, and is the mean of the grouped data. Then the

sample standard deviation s for the grouped data is

8-4 Bernoulli Trials and Binomial Distributions

• Bernoulli Trials A sequence of experiments is called a

sequence of Bernoulli trials, or a binomial experiment, if

1. Only two outcomes are possible on each trial. 2. The probability of success p is the same for each

trial (the probability of failure is then ). 3. All trials are independent.

q = 1 - p s = B(x1 - x)2f₁ + (x₂ - x)2f₂ + p + (x_k - x)2f_k n - 1 x fi xi s2_, s = B(x1 - x)2 + (x₂ - x)2 + p + (x_n - x)2 n - 1 x x1, x2, p , xn  x [mean] = x1f1 + x₂f₂ + p + x_kf_k n

• Binomial Distributions Let the random variable

represent the number of successes in n Bernoulli trials. The probability distribution of called a binomial

distribution, is given by

where p is the probability of success and q is the probabil- ity of failure on each trial.

The mean and standard deviation of a binomial distrib- ution are given by the formulas and

respectively.

8-5 Normal Distributions

Normal probability distributions, or normal curves,

are continuous curves that approximate the relative frequency distributions of measurements such as IQ scores, heights and weights of people, and errors in laboratory experiments.

• Properties of Normal Curves

1. Normal curves are bell-shaped and are symmetri- cal with respect to a vertical line.

2. The mean is at the point where the axis of sym- metry intersects the horizontal axis.

3. The shape of a normal curve is completely determined by its mean and standard deviation

4. The area between a normal curve and the horizontal axis is always 1.

5. 68.3% of the area under a normal curve lies within 1 standard deviation of the mean, 95.4% within 2 standard deviations, and 99.7% within 3 standard deviations.

• Areas under Normal Curves In any normal distribution,

the probability that x lies between a and b is equal to the area under the normal curve from a to b. Such areas can be found from a table of areas associated with the standard normal curve, that is, the normal curve with mean 0 and standard deviation 1. The area under any normal curve from to is equal to the area under the standard normal curve from 0 to z, where

• Approximating Binomial Distributions with Normal Curves To approximate a binomial distribution that is

associated with a sequence of n Bernoulli trials, each having probability of success p, we choose a normal distribution with mean and standard deviation

As a rule of thumb, the normal distribution provides a reasonable approximation if the interval

lies entirely within the interval [0, n]. [ - 3,  + 3]  = 1npq.  = np z =x - . x =  + z  P(a x  b) .    = 1npq,  = np = Cn,xpxqn - x

3. (A) Draw a histogram for the binomial distribution

(B) What are the mean and standard deviation?

4. For the set of sample measurements 1, 1, 2, 2, 2, 3, 3,

4, 4, 5, find the (A) Mean (B) Median (C) Mode

(D) Standard deviation