Business Statistics Chapter 3

The average age of undergraduate students at Grand Canyon University is 44. If the standard deviation is 4, what percentage of undergraduate students are between 36 and 52 years old?

75%

What is the variance of the following sample data? 8, 6, 2, 8

8

given the following weights (in ounces) of four apples, 6, 8, 10, and 7, which of the following is true?

The variance would be in ounces-squared

population standard deviation formula

σ=√(Σ(x−u)^2/N)

Major properties of the median

1. It is not affected by extremely large or small values. Therefore, the median is a valuable measure of location when such values do occur. 2. It can be computed for ordinal-level data or higher. Recall from Chapter 1 that ordinal-level data can be ranked from low to high.

What is the variance of the following sample data? 2, 6, 2, 10

14.67 first find sample mean from xbar = all x / n then the sample variance s^2 = sigma(x-xbar)^2 / n-1

Suppose the sample variance is calculated for a data set containing the ages, in years, of a sample of visitors to the zoo and it is 4. What is the standard deviation?

2

What is the standard deviation of the following population data? 3,1,2,9,5

2.18 ish??

What is the standard deviation of the following population data? 3, 6, 2, 9

2.74 find u -> plug into standard dev formula

What is the variance of the following population data? 2, 6, 1

4.67

What is a parameter?

A characteristic of a population

Weighted Mean

A convenient way to compute the arithmetic mean when there are several observations of the same value. xbar of w=Σ(wx)/Σw the denominator is always the sum of the weights it is a special case of the arithmetic mean

For which of the levels of measurement can the mode be used? select all that apply A. Nominal B. ordinal C. non-skewed D. interval E. ratio

A, B, D, E

The mean, median, and mode are all the same for which type of distribution? A. a symmetrical distribution B. a negatively skewed distribution C. a positively skewed distribution

A. for any symmetric distribution, the mode, median, and mean are located at the center and are always equal

Chebyshev's Theorem

Applies to any set of observations (sample or population), the proportion of the values that lie within k standard deviations of the mean is at least 1-1/k^2, where k is any value greater than 1. at least 3/4 of the values must lie between the mean plus two standard deviations and the mean minus two standard deviations. relationship applies regardless of the shape of the distribution at least 8/9 (88.9%) will lie between plus three standard deviations and minus three standard deviations of the mean. at least 24/25 (96%) will lie between plus and minus five standard deviations of the mean. The distribution of values can have any shape

Which one of the following would be an example of a measure of location? A. A variance B. A range C. A standard deviation D. An average

D. An average

Data levels and measurement

Each of the levels of measurement provides a different level of detail. Nominal provides the least amount of detail, ordinal provides the next highest amount of detail, and interval and ratio provide the most amount of detail.

What does a small value for a measure of dispersion tell us about a set of data?

It indicates that the data are closely clustered around the center

Which of the following is an advantage of the mode?

It is not affected by extreme values

Which of the following are advantages of the variance compared to the range?

It uses all of the values in the data, not just two.

In a certain neighborhood most of the houses cost about $60,000 and the median cost is $76,000. One house cost $700,000 and the mean cost was $82,500. The distribution of housing costs is:

Positively skewed

Most of the items sold at a garage sale cost about $12. Nothing sold for more than $15, but a few items cost only a few cents. Which of the following would NOT be a good measure of the center of the distribution of sale prices? select all that apply

Standard deviation and mean

What is the median?

The midpoint of the values after they have been ordered from the minimum to the maximum values. Describes data better than the mean. Data must be at lease an ordinal level of measurement.

What characteristic of a data set makes the median the better measure of the center of the data than the mean?

When the data set included one or two very large or very small values

What is a statistic?

a characteristic of a sample

which of the following are true regarding the variance?

it measure squared distance from the mean

What is dispersion?

often called the variation or the spread in the data. To describe it, we consider the range, the variance, and the standard deviation. The range is the simplest measure.

What is a measure of location and what is the purpose?

often referred to as averages. the purpose is to pinpoint the center of a distribution of data

What is variance?

the arithmetic mean of the squared deviations from the mean. For populations whose values are near the mean, the variance will be small. For populations whose values are dispersed from the mean, the population variance will be large. o^2=Σ(x−u)^2/N o^2 - population variance x - the value of a particular observation in the population u - the arithmetic mean of the population N - the number of observations in the population

The larger the population variance is for a data set,

the more spread out the data is

sample standard deviation is:

the square root of sample variance

The population standard deviation is

the square root of the population variance

Sample mean and formula

the sum of all the sampled values divided by the total number of sampled values. Xbar =Σx/n

Population mean and formula

the sum of all the values in the population divided by the number of values in the population. The arithmetic mean of the values in the population. u=Σx/N

Which of the following is true regarding medians and means? A. Medians can be calculated from ordinal-level data, but means can't B. Medians and means cannot be calculated from interval-level data. C. Medians and means can be calculated from nominal-level data D. Means can be calculated from ratio-level data, but medians can't

A.

Which of the following kinds of data can be used to find the median value? select all that apply A. Interval level data B. ordinal level data C. ratio level data D. nominal level data

A. , B. , and C.

The median would be a better measure for the center of the data for of which of the following data sets? A. {11, 14, 16, 1, 12, 15, 17} B. {4, 6, 5, 8, 4, 7} C. {6, 6, 7, 7, 8, 8, 9} D. {1.2, 1.4, 1.6, 1.1, 1.3}

A. {11, 14, 16, 1, 12, 15, 17} The value 1 is much smaller than the rest of the data. Best when there are very large or small values in the data set

what is the range?

the difference between the largest and the smallest values in a data set

Empirical Rule

Also called Normal Rule, it describes relationships involving the standard deviation and the mean. For a symmetrical, bell-shaped distribution, this allows more precision in explaining the dispersion about the mean. For a symmetrical, bell-shaped frequency distribution, approximately 68% of the observations lie within plus and minus one standard deviation of the mean; about 95% of the observations will lie within plus and minus two standard deviations of the mean; and practically all (99.7%) will lie within plus and minus three standard deviations of the mean.

which of the following are reasons why the mode would NOT be the good choice of measure for the describing the center of a set of data? check all that apply

the data may be bimodal (it would be difficult to describe the center of the data set with two differnt values) there may be no observation that occurs more than once (in this case there would be no mode) the most frequent observation is much higher or much lower than most of the data values. (in this case, the mode wouldn't be very representative)

How does the formula for the sample mean differ from the formula for population mean?

the formulas are functionally the same, but 'n' (the sample size) is used instead of 'N' (the population size) The greek letter mu, u, is used to represent the population while, x-bar is used for the sample mean

According the the empirical rule, a certain percentage of observations will be found within a specific number of standard deviations of the mean for a normal distribution. match the percentage of observations to the number of standard deviations.

68% Plus or minus one standard deviation 95% Plus or minus two standard deviations 99.7% Plus or minus three standard deviations

Sample variance formula

s2=Σ(x−xbar)^2/n−1

What is the standard deviation of the following sample data? 7,6,2,0,5

2.92

Which of the following statements are true of the weighted mean? select all that apply A. The denominator of the weighted mean is always the sum of the weights B. it is not unduly influenced by very large or very small values C. It is used with data that has repeated values, such as a frequency distribution D. it gives a smaller value for the mean than the formula for the arithmetic mean

A, C

Properties of the Arithmetic Mean

The arithmetic mean is a widely used measure of location. It has several important properties: 1. To compute a mean, the data must be measured at the interval or ratio level. Recall from Chapter 1 that ratio-level data include such data as ages, incomes, and weights. 2. All the values are included in computing the mean. 3. The mean is unique. That is, there is only one mean in a set of data. Later in the chapter, we will discover a measure of location that might appear twice, or more than twice, in a set of data. 4. The sum of the deviations of each value from the mean is zero. Expressed symbolically: Σ(x−xbar)=0

Which of the following are used to measure dispersion? Check all that apply. A. average B. range C. mean D. standard deviation E. median F. variance

B. range D. standard deviation F. variance

Which of the following is true regarding the application of Chebyshev's theorem and the Empirical Rule?

The Empirical Rule predicts a higher percentage than Chebsyshev's Theorem the empirical rule gives more precise answers for the symmetrical, bell-shaped distribution (empirical rule works for any distribution) Chebsyshev's theorem works for symmetrical, bell-shaped distributions (and other distributions)

Which of the following statements describe the weaknesses of the range as a measure of dispersion?

1. It may be unduly influenced by an especially small value. 2. It may be unduly influenced by an especially large value. 3. Only two values from the data set are used