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1. Eighteen SJSU students were surveyed about their opinions on two teaching evaluations: ratemyprofessor.com and SOTES (Student Opinion of Teaching Effectiveness). They indicated the level of effectiveness on a 7-point scale, with “1” being “not very effective” and “7” being “very effective”. The rating scores are shown below by teaching evaluation.Observation SOTES ratemyprofessor.com1 4 72 6 63 5 44 6 55 7 36 5 47 7 68 4 29 510 3A) Which teaching evaluation is more effective? Show all the calculations (1 pt).B) Students have more different or diverse opinions on which teaching evaluation? Use all the observations and show all the calculations (3 pts).2. Ms. Green is responsible for recruiting international students for a summer internship at her company. Her recruiting job is almost done, but she needs to pick one last student from three candidates. They are all very competitive and it seems that the only way to rank them is their college GPAs. Ming is from China and his score is 91.8 out of 100. Leif is from Norway and his score is 4.21 out of 5. Jason is from the U.S. and his score is 3.55 out of 4. Luckily, Ms. Green also has the mean and standard deviation of college GPAs from each country (see table). If you were Ms. Green, which candidate will you choose for the summer internship? Show all the calculations (2 pts).Country Mean SD Candidate’s Score China 82.7 8.9 91.8 (Ming) Norway 2.46 1.63 4.21 (Leif) US 2.24 1.29 3.55 (Jason) 3.Suppose a fast-food restaurant wishes to estimate average sales volume for a new menu item. The restaurant has analyzed the sales of the item at a similar outlet and observed the following results:X = 715 (mean daily sales)S = 114 (standard deviation of sample)n = 36 (sample size)The restaurant manager wants to know into what range the mean daily sales should fall 95 percent of the time. Perform this calculation (2 pts).4. A group of marketing researchers study the expenditure on dinning out. They want to have a 95 percent confident level (Z) and accept a magnitude of error (E) of less than \$2.75. The estimate of the standard deviation is \$15.80 based on their pilot study. What is the calculated sample size if they want to run a survey? Perform this calculation (2 pts)
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Ch 13
Big Data Basics: Describing
Samples and Populations
Dr. Jing Zhang
BUS 138 Marketing Research
LEARNING OUTCOMES
1. Use basic descriptive statistics to analyze
data and make basic inferences about
population metrics.
2. Distinguish among the concepts of population,
sample, and sampling distributions
3. Explain the central-limit theorem
4. Use confidence intervals to express inferences
5. Understand the major issues in specifying
sample size.
13–2
Introduction
• All the statistics in this chapter are univariate in
the sense that only one variable is involved
• This chapter provides a good background for
understanding equations related to sample size
requirements
13–
3
Descriptive Statistics and
Basic Inferences
• Raw data are simply numbers and words – with little
meaning
• The most basic statistical tools for summarizing
information from data include:
• Frequency distributions
• Proportions
• Measures of central tendency and dispersion
• Metrics provide a means of comparison
13–
4
• Metrics: A summary number that allows analysts to
compare the characteristics of a sample with some
population benchmark, characteristics of another
sample, or some other critical values.
• Inferential statistics: A summary representation of data
from a sample that allows us to understand (i.e., infer
from sample to population) an entire population.
• Two applications of statistics:
• To describe characteristics of the population or sample
and
• To generalize from a sample to a population
13–5
Descriptive Statistics and
Basic Inferences (cont’d.)
• Sample statistics: Summary measures about variables
computed using only data taken from a sample.
• English letters to denote sample statistics (e.g., X or
S)
• Population parameters: Summary characteristics of
information describing the properties of a population.
• Greek lowercase letters to denote population
parameters (e.g.,  or )
13–6
What Are Sample Statistics and
Population Parameters?
Frequency Distributions
• Constructing a frequency table or frequency
distribution is one of the most common means of
summarizing a set of data
• The frequency of a value is the number of times a
particular value of a variable occurs
• A distribution of relative frequency, or a percentage
distribution, is developed by dividing the frequency
of each value by the total number of observations,
and multiplying the result by 100
• Probability is the long-run relative frequency with
which an event will occur
13–
7
13–8
EXHIBIT 13.1 Frequency Distribution of Deposits
EXHIBIT 13.2 Percentage Distribution of Deposits
13–9
Relative Frequency
EXHIBIT 13.3 Probability Distribution of Deposits
13–10
Long-term relative frequency
Proportions
• A proportion indicates the percentage of
population elements that meet some criteria for
membership in a category.
• May be expressed as a percentage, a fraction, or
a decimal number
13–
11
Top-Box/Bottom-Box Scores
• A top box score generally refers to the portion of
respondents who choose the most favorable choice in a
multiple-choice question usually dealing with customer
opinions.
• The portion that would highly recommend a business to
others or the portion expressing the highest likelihood of
• The logic is that respondents who choose the most
extreme response are really quite unique compared to
the others
13–
12
Top-Box/Bottom-Box Scores (cont’d.)
Percentage of participants by agreement level:
“I will highly recommend Kindle to my friends.”
30
25
What is the
top box score?
20
15
%
10
5
0
SD
D
N
A
SA
13
Top-Box/Bottom-Box Scores (cont’d.)
• Managers should examine the bottom-box score –
the portion of respondents who choose the least
favorable response to some question about
customer opinion
• More diagnostic of customer problems
• Often signals a need for some managerial reaction
• Net Promoter Score (NPS) reveals the frequency of
promoters (top-box) and distractors (bottom-box).
13–
14
Central Tendency Metrics: The Mean

Sample Mean
13–
15
Central Tendency Metrics: The Median
• The midpoint of the distribution, or the 50th
percentile
• The value below which half the values in the sample
fall
• A better measure of central tendency in the
presence of extreme values or outliers
13–
16
Central Tendency Metrics: The Mode
• The measure of central tendency that merely
identifies the value that occurs most often
• Determined by listing each possible value and noting
the number of times each value occurs
• Used for data that is less than interval, with one
large peak
13–
17
Dispersion Metrics
• Accurate analysis of data also requires knowing the
tendency of observations to depart from the central
tendency
• Another way to summarize the data is to calculate
the dispersion of the data, or how the observations
vary from the mean
13–
18
13–19
EXHIBIT 13.5 Sales Levels for Two Products with Identical Average Sales
Dispersion Metrics: The Range
• The simplest measure of dispersion – the distance
between the smallest and largest values of a
frequency distribution
• Does not take into account all the observations
• Indicates the extreme values of the distribution
• In a skinny distribution, values are a short distance
from the mean; in a fat distribution values are
spread out (see Exhibit 13.6 on the next slide).
13–
20
13–21
EXHIBIT 13.6 Low Dispersion versus High Dispersion
Dispersion Metrics: Deviation Scores
• A deviation of any observation from the mean can
be calculated by subtracting the mean from that
observation
• In Exhibit 13.5: January Product A: d= 196 – 200 = -4;
Product B: d=150-200 = -50
13–
22
Dispersion Metrics:
Why Use the Standard Deviation?
• It is perhaps the most valuable index of spread, or
dispersion
• Variance – useful for describing the sample
variability; will equal to zero if and only if each and
every observation in the distribution is the same as
the mean
13–
23
Why Use The Standard Deviation?
(cont’d.)
• Standard deviation
• The square root of the variance for distribution is
called the standard deviation
• It is in the original measurement units (e.g., \$)
rather than in squared units (e.g., \$^2)
• S is the symbol for the sample standard deviation
Standard Deviation =
13–
24
13–25
EXHIBIT 13.7
Calculating a Standard Deviation: Number of Sales Calls per
Day for Eight Salespeople
Video: How to Calculate Standard Deviation?
LEARNING OUTCOMES
1. Use basic descriptive statistics to analyze data
and make basic inferences about population
metrics.
2. Distinguish among the concepts of
population, sample, and sampling
distributions
3. Explain the central-limit theorem
4. Use confidence intervals to express inferences
5. Understand the major issues in specifying
sample size.
13–26
The Normal Distribution
• Normal Distribution
• A symmetrical, bell-shaped distribution (normal
curve) that describes the expected probability
distribution of many chance occurrences.
• IQ scores, SAT scores, shoe size, quarterly
revenue.
• Standardized Normal Distribution
• A purely theoretical probability distribution that
reflects a specific normal curve for the
standardized value, z.
13–27
EXHIBIT 13.8 Normal Distribution: Distribution of
Intelligence Quotient (IQ) Scores
• The graph is symmetrical about the mean.
• Curve is always above horizontal axis.
• Mean, mode, and median are converged on the
same point.
13–28
EXHIBIT 13.8 Normal Distribution: Distribution of Intelligence
Quotient (IQ) Scores

The total area under the curve equals 100%.
68.3% = + – 1 SD of the mean. [xx, xxx]
95.4% = + – 2 SD’s of the mean. [xx, xxx]
99.7% = + – 3 SD’s of the mean. [xx, xxx]
13–29
EXHIBIT 13.9
Standardized Normal Distribution
13–30
The Normal Distribution (cont’d)
• Characteristics of a Standardized Normal Distribution
1. It is symmetrical about its mean.
2. The mean identifies the normal curve’s highest
point (the mode) and the vertical line about which
this normal curve is symmetrical.
3. The normal curve has an infinite number of cases
(it is a continuous distribution), and the area under
the curve has a probability density equal to 1.0.
4. The standardized normal distribution has a mean
of 0 and a standard deviation of 1.
13–31
Standardized Normal Table: Area under Half of the Normal Curve
13–32
EXHIBIT 13.10
Http://www.mathisfun.com/data/standard-normal-distribution-table.html
• The standardized normal distribution is extremely
valuable because we can translate or transform any
normal variable, X, into the standardized value, Z
• This has many pragmatic implications for the
marketing researcher
• A typical standardized normal table allows us to
evaluate the probability of the occurrence of certain
events without any difficulty
13–33
The Standardized Normal
Distribution and Z-Scores
Computing Z Scores
• Subtract the mean from the value to be
transformed, and divide by the standard deviation
(all expressed in original units)
• In the formula, note that σ, the population standard
deviation, is used for calculation:
X −
Z =

where μ is the hypothesized or expected value of the
mean
13–
34
Population Distribution and
Sample Distribution
13–35

Three Important Distributions
13–36
Sampling Distribution (cont’d.)

13–
37
Sampling Distribution
• Defined as a theoretical probability distribution that
shows the functional relation between the possible
values of some summary characteristic of n cases
drawn at random and the probability associated
with each value over all possible samples of size n
from a particular population.
• In a nutshell, sampling distribution is a portray of the
means of all possible samples of a given size.
13–
38
Sampling Distribution: Example
• Study the dollar amount that kids spend on their
most recent toy in the U.S.
• Randomly choose a sample of 20 kids and ask how
much they spend on the recent toy.
• I then randomly sample another 20 kids and record
the same information.
• I do this a total of 6 times. The results are displayed
in the table on the next slide.
13–
39
Sampling Distribution: Example (Cont’d)
Sample ID
Sample Size
Average of Dollar Amount
#1
20
26.8
#2
20
25.4
#3
20
27.5
#4
20
32.6
#5
20
30.1
#6
20
23.8
• Each sample has its own mean value, and each value is different
• Continue this experiment by selecting and measuring more samples
and observe the pattern of sample means
• This pattern of sample means represents the sampling distribution for
the dollar amount kids spend on toys.
13–
40
Sampling Distribution: Example (Cont’d)
• What happens to the sampling distribution if we
increase the sample size?
• As the sample size (n) gets larger, the sample means
tend to follow a normal probability distribution; they
tend to cluster around the true population mean.
Hence, the sampling distribution approaches to a
normal distribution.
13–
41
LEARNING OUTCOMES
1. Use basic descriptive statistics to analyze data
and make basic inferences about population
metrics.
2. Distinguish among the concepts of population,
sample, and sampling distributions
3. Explain the central-limit theorem
4. Use confidence intervals to express inferences
5. Understand the major issues in specifying
sample size.
13–42
Central Limit Theorem

13–
43
13–44
EXHIBIT 13.13
The Mean Distribution of Any Distribution Approaches Normal as n
Increases
13–45
EXHIBIT 13.13
The Mean Distribution of Any Distribution Approaches Normal as n
Increases (cont’d.)
LEARNING OUTCOMES
1. Use basic descriptive statistics to analyze data
and make basic inferences about population
metrics.
2. Distinguish among the concepts of population,
sample, and sampling distributions
3. Explain the central-limit theorem
4. Use confidence intervals to express
5. Understand the major issues in specifying
sample size.
13–46
Estimation of Parameters:
Point Estimate
13–47

Estimation of Parameters:
Confidence Intervals

13–
48
Calculating a Confidence Interval
Approximate location (value) of the population mean
Estimation of the sampling error
13–49
Step By Step Calculation of the
Confidence Interval
13–50
LEARNING OUTCOMES
1. Use basic descriptive statistics to analyze data
and make basic inferences about population
metrics.
2. Distinguish among the concepts of population,
sample, and sampling distributions
3. Explain the central-limit theorem
4. Use confidence intervals to express inferences
5. Understand the major issues in specifying
sample size.
13–51
• Three factors required to specify sample size
• 1. The variance, or heterogeneity, of the population
in statistical terms refers to the standard deviation of
the population parameter
• A heterogeneous population has more variance (a
larger standard deviation) which will require a
larger sample.
• A homogeneous population has less variance (a
smaller standard deviation) which permits a
smaller sample.
13–52
Factors in Determining Sample Size
for Questions Involving Means
• 2. The magnitude of error, or the confidence
interval, is defined in statistical terms as E
• How precise must the estimate be?
• From a managerial perspective, the importance of
the decision in terms of profitability will influence
the researcher’s specifications of the range of
error
• 3. Confidence level (typically 95 percent)
• How much error will be tolerated?
13–53
Factors in Determining Sample Size for
Questions Involving Means (cont’d.)
13–54
EXHIBIT 13.16 Statistical Information Needed to Determine Sample Size for
Questions Involving Means
Estimating Sample Size for Questions
Involving Means
• Estimating sample size:
13–55
Sample Size Example
• Suppose a survey researcher, studying
expenditures on lipstick, wishes to have a 95
percent confident level (Z) and a range of error (E)
of less than \$2.00. The estimate of the standard
deviation is \$29.00. What is the calculated sample
size?
13–56
Sample Size Example (Cont’d)
• Suppose, in the same example as the one before, the range
of error (E) is acceptable at \$4.00. Sample size is reduced.
• Doubling the range of acceptable error reduces sample size
requirement dramatically.
13–57
• Sample size may also be determined on the basis of
managerial judgments
• Using a sample size similar to those used in
previous studies.
• Another judgmental factor is the selection of the
appropriate item, question, or characteristics to
be used for the sample size calculations
• Often the item that will produce the largest
sample size will be used to determine the ultimate
sample size
13–58
Determining Sample Size on the
Basis of Judgment
• Another consideration stems from most researchers’
need to analyze the various subgroups within the
sample
• Rule of thumb for selecting minimum subgroup
sample size: each subgroup to be separately
analyzed should have a minimum of 100 or more
units in each category of the major breakdowns
13–59
Determining Sample Size on the
Basis of Judgment (cont’d.)