OPRE504
Chapter Study Guide
Chapter 10
I.
Hypothesis Test for Proportions
One-proportion z-Test
1.
State Hypotheses
H0: P = P0
Ha: P ≠ P0 [ two-tailed test]
H0: P = P0
Ha: P > P0 [ one-tailed test, upper]
H0: P = P0
Ha: P < P0 [ one-tailed test, lower]
2.
Calculate Standard Deviation of the Sampling Distribution based on H0
SD (𝑝̂ ) = √
3.
𝑝0𝑞0
𝑛
(q0 = 1-p0)
Calculate Z-statistic and find probability associated with Z using Z-Table [A32/33]
𝑝̂− 𝑝0
Z = 𝑆𝐷(𝑝̂ ) , using Z-table find out P-value [ p-value is defined as the probability of
observing the sample value (proportion) given the null hypothesis]. For two-tailed test, p-value
is doubled since Z-table only gives one-tailed probability. [Refer to Sharpe 2011, Figure 10.3
and 10.4 on pp.294-295]
4.
Decision
Reject H0 if p-value is very small, indicating the observed value is very unlikely if the
null hypothesis is true;
When using 0.1%, 1%, 5%, and 10% as threshold P-values, we call them alpha levels (or
significance levels). If calculated probability is less than stated alpha level, we reject H0.
Q10.1 [Sharpe 2011, Ch.10, Exercise 23] The National Center for Education Statistics monitors
many aspects of elementary and secondary education nationwide. Their 1996 numbers are often
used as a baseline to assess changes. In 1996, 34% of students had not been absent from school
even once during the previous month. In the 2000 survey, responses from 8302 students showed
that this figure had slipped to 33%. Officials would be concerned if student attendance were
declining. Do these figures give evidence of a decrease in student attendance?
Step 1.
State Hypotheses
Step 2.
SD =
Step 3.
z =,
p-value =
Chaodong Han
OPRE504
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Step 4.
Decision
Q10.3 A candy company claims that in a large bag of holiday M&M half the candies are red and
half the candies are green. You pick candies at random from a bag and discover that of the first
20 you eat, 12 are red. [Sharpe 2011, Chapter10, Exercise 11, p.315]
a)
If it were true that half are red and half are green, what is the probability you would
have found that at least 12 out of 20 were red?
SD(p) =
Z=
Check Z-table, or Excel formula, p-value = NORMSDIST(z) =
b).
Do you think that half of the M&M’s candies in the bag are really red at the 10%
significance level? Explain.
Step 1. State Hypotheses:
Step 2.
Step 3. Z =, one-tailed p-value =
; two-tailed is
Step 4. Decision
More Exercises:
1.
Guided Example – Foreclosure [Sharpe 2011, pp.257-8]
2.
Textbook [Sharpe 2011] Chapter 10, Exercises 18, 19, 20, 21, 22, 24, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, and 42.
Chaodong Han
OPRE504
Page 2 of 2