1. Statistics 200 Objectives:
Lecture #15 Tuesday, October 11, 2016
Textbook: Sections 8.7, 9.1 through 9.4
Objectives:
• Approximate binomial-distribution probabilities using a normal distribution
• Explain statistical inference in terms of statistics and parameters.
• Define the sampling distribution of a statistic.
• Describe the sampling distribution of a sample proportion.

2. Binomial distributions
n fixed at 10,
p increasing
p fixed at 0.02,
n increasing

3. Binomial Distribution Example
Studies have shown that about 57% of college graduates
are women. A representative sample of n =100 college
graduates was obtained.
Let X = number in the sample who are women.
Then X is basically a binomial random variable.
1. What is the expected number of college graduates that are women?
2. What is the value of the standard deviation for this binomial distribution?
3. Does this binomial distribution look normal?

4. Binomial Distribution example cont’d
Binomial distribution function looks normal!
Idealized Range for x:
Actual Range for x:
0, 1, … , n
about 42 to 72

5. When should we keep binomials discrete?
Guideline: We can use the normal approximation when n×p and n×(1-p) are both at least 10.

6. Back to our binomial distribution example
In this case, n=100 and p= So both np and n(1-p) are larger than 10.
Exact P(X<60):
0.692
Let’s try using normal approximation.
(Think about why!)

7. Back to our binomial distribution example
In this case, n=100 and p= So both np and n(1-p) are larger than 10.
Approximate P(X<60):
First step is to convert to a z-score.

8. Back to our binomial distribution example
Approximate one more time but using the continuity correction:
Approximate P(X<60):
First step is to convert to a z-score.

9. Back to our binomial distribution example
Approximate P(X<60):
• Exact value:
0.692
• Normal approx:
• Normal approx with c.c.:
0.693

10. We now begin a strong focus on Inference
Means
Proportions
One population mean
One population proportion
Two population proportions
Difference between Means
Mean difference
This week

11. The value of one of these may be different from sample to sample.
parameter
Number that describes a population: _____________
Number that describes a sample:
statistic or ______ _______
sample estimate.
The value of one of these may be different from sample to sample.
Which one?
To describe the possible values, use the
__________ ___________
sampling distribution

12. Sampling distribution
…basically, the possible values of the sample statistic.
The sampling distribution for a statistic is the probability distribution of possible values of the statistic for repeated samples of tehs aem size taken from the same population.

13. Motivation
Eventual Goal: Use statistical inference to answer the question “Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%?”
Strategy:
Get a random sample of 90 individuals and ask them this question. Use the answers to perform a hypothesis test to answer the question.

14. Let X = # who prefer chocolate in our sample
binomial
underlying distribution of X: ___________
type of data: categorical quantitative
parameter: population proportion
statistic: sample proportion
where:
X = number of successes
n = sample size

15. Another Example
Historically: 70% of PSU students list Penn State as their first choice.
Let X = number of PSU students who said Penn State was their first choice in a sample where n = 100
Sampling: repeated samples of n = 100 are taken from the PSU population
Population
p = 0.70
(fixed)
Sample? x?
Sample1 x1
Sample4 x4
Sample2 x2
Sample3 x3

16. Results of 5,000 random samples from population (n = 100 & p = 0.70)
X = # people who had PSU as first choice.
Binomial with
n=100
p=0.70

17. Results of 5,000 random samples from population (n = 100 & p = 0.70)
For each sample, calculate:
What does the distribution of the p-hat values look like?

18. Binomial Distribution vs. p-hat sampling distribution: n = 100 & p = 0
mean = np = 70
mean = p = 0.70
4.58

19. If you understand today’s lecture…
8.81a, 8.83, 8.85 (for the normal distribution problems, sketch a picture!), 9.1, 9.3, 9.7, 9.9, 9.11, 9.33, 9.35, 9.37
Objectives:
• Approximate binomial-distribution probabilities using a normal distribution
• Explain statistical inference in terms of statistics and parameters.
• Define the sampling distribution of a statistic.
• Describe the sampling distribution of a sample proportion.