CORE COMPETENCIES
1. Define/discuss following terms: population, sample, parameter, statistic,
descriptive statistics (graphical and numeric), inferential statistics (estimation and
hypothesis testing), random variable, data, interval data, nominal data, ordinal
data, time series data, cross sectional data.
2. For Following Nominal (qualitative) Data [red, blue, red, red, white] Complete:
frequency/relative frequency tables, bar charts, pie charts.
3. For Following Interval (numeric) Data [.11, .98, 1.55, 2.80, 3.34, 5.47, 5.55, 5.89,
6.66, 8.22] Complete: frequency/relative frequency table, histograms (shapes –
symmetric, skewed, unimodal, bimodal???), ogive (cumulative frequency
histogram).
4. Make up a numeric example for the following two statistical tools: contingency
table, scatter diagram.
5. Numeric Techniques: measures of central location (tendency) – mean, median,
mode. Calculate for following data [1, 5, 7, 6, 7, 3, 5, 7, 6, 3]
6. Numeric Techniques: measures of variability – range, standard deviation,
variance, coefficient of variation. Calculate for following data [1, 5, 7, 6, 7, 3, 5,
7, 6, 3]
7. Numeric Techniques: measures of relative standing – 90th percentile, first, second
and third quartile, interquartile range. Calculate for following data [1, 5, 7, 6, 7, 3,
5, 7, 6, 3]
8. Numeric Techniques: measures of linear relationship – covariance, correlation,
least squares line [will delay until we cover regression]
9. Binomial: Given a binomial random variable with n = 20 and p = .2 find the
following probabilities
a. P(X > 5) =
b. P(X > 5) =
c. P(X < 4) =
d. P(3 < X < 7) =
e. P(X = 5) =
10. Poisson: Given a poisson random variable with  = 4.5, find following
probabilities
a. P(X > 9) =
b. P(X > 9) =
c. P(X = 7) =
d. P(4 < X < 6) =
1. Normal Distribution: Given a random variable X (time to receive service at
BurgerLand) that is normally distributed with mean μ = 5 minutes and standard
deviation σ = .8 minutes, find the following probabilities for the next arrival.
a. P(X > 6) =
b. P(X > 6) =
c. P(X < 6) =
d. P(X < 3) =
e. P(3 < X < 4) =
f. Assuming the standard deviation would not change, what would the mean
μ need to be if you wanted to be able to service 99% of your customers in
less than 6 minutes?
2. Sampling Distribution of Sample Mean: Given a random variable X (time to
receive service at BurgerLand) with mean μ = 5 minutes and standard deviation σ
= .8 minutes, a random sample of 100 (n) arrivals was taken and the sample mean
(X-bar) computed.
a. What is the sampling distribution of the sample mean (X-bar)
b. What assumptions must you make about your answer in a. above.
c. For a. above what is the probability that the sample mean (X-bar) is
greater than 5.5 minutes?
d. If the sample size were n = 9 instead of n = 100, what is the sampling
distribution of the sample mean (X-bar)
e. What assumptions must you make about your answer in c. above.
f. For d. above what is the probability that the sample mean (X-bar) is
greater than 5.5 minutes?
3. Sampling Distribution of the Proportion: Given a random variable ps (the sample
proportion of customers who are satisfied with their service at BurgerLand) where
p = .8 (true proportion who are satisfied), a random sample of 100 (n) arrivals was
taken and the sample proportion (ps) computed
a. What is the sampling distribution of the sample proportion ps
b. What assumptions must you make about your answer in a. above.
c. If the sample size were n = 9 instead of n = 100, what is the sampling
distribution of the sample proportion ps
d. What assumptions must you make about your answer in c. above.
4. Single Mean Confidence Intervals (Large Sample): Given a random variable X
(time to receive service at BurgerLand) with mean μ unknown and standard
deviation σ = .8 minutes, a random sample of 100 (n) arrivals was taken and the
sample mean (X-bar) computed to be 4.6 minutes and sample standard deviation
computed to be .7
a. Provide a point estimate for the true mean service time (μ)
b. Provide a 95% confidence interval estimate for the true mean service time
(μ).
c. Provide a 99% confidence interval estimate for the true mean service time
(μ).
5. Single Mean Confidence Intervals (Large Sample): Given a random variable X
(time to receive service at BurgerLand) with mean μ unknown and standard
deviation σ unknown, a random sample of 100 (n) arrivals was taken and the
sample mean (X-bar) computed to be 4.6 minutes and sample standard deviation
computed to be .7
a. Provide a point estimate for the true mean service time (μ)
b. Provide a 95% confidence interval estimate for the true mean service time
(μ).
c. Provide a 99% confidence interval estimate for the true mean service time
(μ).
6. Single Mean Confidence Intervals (Small Sample): Given a random variable X
(time to receive service at BurgerLand) with mean μ unknown and standard
deviation σ = .8 minutes, a random sample of 9 (n) arrivals was taken and the
sample mean (X-bar) computed to be 4.6 minutes and sample standard deviation
computed to be .7
a. Provide a point estimate for the true mean service time (μ)
b. Provide a 95% confidence interval estimate for the true mean service time
(μ).
c. Provide a 99% confidence interval estimate for the true mean service time
(μ).
d. What assumption must you make to work this problem?
7. Single Mean Confidence Intervals (Small Sample): Given a random variable X
(time to receive service at BurgerLand) with mean μ unknown and standard
deviation σ unknown, a random sample of 9 (n) arrivals was taken and the sample
mean (X-bar) computed to be 4.6 minutes and sample standard deviation
computed to be .7
a. Provide a point estimate for the true mean service time (μ)
b. Provide a 95% confidence interval estimate for the true mean service time
(μ).
c. Provide a 99% confidence interval estimate for the true mean service time
(μ).
d. What assumption must you make to work this problem?
8. Sample Size to Estimate the Mean: Given a random variable X (time to receive
service at BurgerLand) with mean μ unknown and standard deviation σ = .8
minutes, how large a sample should you take in order to estimate the mean within
+ e (.1 minutes) with 95% confidence?
9. Confidence Interval Estimate for the True Proportion (large sample): A random
sample of 100 BurgerLand customers was asked if they were satisfied with their
service. 80 responded yes and 20 responded no.
a. Provide a point estimate for the true proportion who are satisfied?
b. Provide a 95% confidence interval estimate for the true proportion who
are satisfied.
c. Provide a 99% confidence interval estimate for the true proportion who
are satisfied.
d. What assumption must you make to work this problem?
e. Answer parts a through d above if the sample size was 9 and 6 responded
yes (Small Sample).
CORE COMPETENCIES – Single Mean & Proportion Hypothesis Test
10. Understand concept of Type I and Type II errors and the probabilities of
committing these errors (α and β) as they relate to the hypothesis testing process.
11. Single Mean Hypothesis Test (Large Sample, std. dev. known): Given a random
variable X (time to receive service at BurgerLand) with mean μ and standard
deviation σ = .8 minutes, a random sample of 100 (n) arrivals was taken and the
sample mean (X-bar) computed to be 4.6 minutes and sample standard deviation
computed to be .7
a. The store’s manager claims that the average time at his store to receive
service is 3 minutes, but you suspect that he just pulled that number out of
the air and the true mean may be larger or smaller that 3 minutes. Test the
hypothesis that the mean is equal to 3 at a 5% level of significance.
b. Sketch the power curve for this hypothesis test.
c. What is the p-value
12. Single Mean Hypothesis Test (Large Sample, std. dev. unknown): Given a
random variable X (time to receive service at BurgerLand) with mean μ unknown
and standard deviation σ unknown, a random sample of 100 (n) arrivals was taken
and the sample mean (X-bar) computed to be 4.6 minutes and sample standard
deviation computed to be .7
a. The store’s manager claims that the average time at his store to receive
service is 3 minutes, but you suspect that he just pulled that number out of
the air and the true mean may be larger or smaller that 3 minutes. Test the
hypothesis that the mean is equal to 3 at a 5% level of significance.
b. Sketch the power curve for this hypothesis test.
c. What is the p-value?
13. Single Mean Hypothesis Test (Small Sample, std. dev. known): Given a random
variable X (time to receive service at BurgerLand) with mean μ unknown and
standard deviation σ = .8 minutes, a random sample of 9 (n) arrivals was taken
and the sample mean (X-bar) computed to be 4.6 minutes and sample standard
deviation computed to be .7
a. The store’s manager claims that the average time at his store to receive
service is 3 minutes, but you suspect that he just pulled that number out of
the air and the true mean may be larger or smaller that 3 minutes. Test the
hypothesis that the mean is equal to 3 at a 5% level of significance.
b. Sketch the power curve for this hypothesis test.
c. What is the p-value?
d. What assumption must you make to work this problem?
14. Single Mean Hypothesis Test (Small Sample, std. dev. unknown): Given a
random variable X (time to receive service at BurgerLand) with mean μ unknown
and standard deviation σ unknown, a random sample of 9 (n) arrivals was taken
and the sample mean (X-bar) computed to be 4.6 minutes and sample standard
deviation computed to be .7
a. The store’s manager claims that the average time at his store to receive
service is 3 minutes, but you suspect that he just pulled that number out of
the air and the true mean may be larger or smaller that 3 minutes. Test the
hypothesis that the mean is equal to 3 at a 5% level of significance.
b. Sketch the power curve for this hypothesis test.
c. What is the p-value?
d. What assumption must you make to work this problem?
15. Sample Size to Achieve Specified Power: Given a random variable X (time to
receive service at BurgerLand) with mean μ unknown and standard deviation σ =
.8 minutes. The store’s manager claims that the average time at his store to
receive service is 3 minutes, but you suspect that he just pulled that number out of
the air and the true mean may be larger or smaller that 3 minutes. To test the
hypothesis that the mean is equal to 3 at a 5% level of significance how large a
sample should you take in order to have a 95% probability of rejecting the null
hypothesis if the true mean were actually 4.
16. Single Proportion Hypothesis Test (large sample): A random sample of 100
BurgerLand customers was asked if they were satisfied with their service. 80
responded yes and 20 responded no.
a. A local famous person (paid commercial) stated on the radio that 90% of
BurgerLand customers are satisfied with their service. You question that
claim and decide to test the hypothesis that the true proportion is equal to
.90 (as the famous person claimed) at a 5% level of significance. Conduct
this hypothesis test.
b. What is the p-value?
17. Set up and distinguish the difference in a one-tailed hypothesis test and a two
tailed hypothesis test.
a. Ho : μ = 4
H1 : μ not= 4
b. Ho : μ < 4
H1 : μ > 4
c. Ho : μ > 4
H1 : μ < 4