10/12/10 Week 8
Confidence Intervals one proportion and one mean
Primary purpose of C.I. is to estimate same unknown population parameter.
Consider the following four research questions on PSU University Park undergraduate students.
Assume a class survey represents random sample of PSU UP undergraduates.
1. Do you think marijuana should be legalized? “Yes, no” Categorical p hat estimates true
2. Do you agree with same sex marriages? “Yes, no” Categorical p hat estimates true
3. What is the mean GPA of PSU UP undergraduate? “number” Quantitative X bar estimating
true mean M.
4. What is the mean amount of money students spend on books in a semester? “Number”
Quantitative X bar estimating true mean M.
Construct confidence intervals for those unknown parameters of P and M.
Confidence Interval Formula:
Sample statistic plus or minus Multiplier times Standard Error (Margin of Error)
Confidence interval for one proportion: p hat + Z X √P(1P)/n
Assuming np hat and n (1p hat) > 5
Ex. # successes and # of failures 25
Confidence interval for one mean
X bar + t X s/√n this “t” comes from table A2 where we consider degrees of freedom, df= n1 S= standard deviation of the variable in our sample ex. It is the sample standard deviation.
There are four common or typical levels of confidence
90% 95% 98% 99%
To apply this interval the population data needs to be approximately normal OR sample size n
needs to be at least 30.
Z multipliers for proportion confidence interval
Proportion (categorical) Questions:
95% confidence interval for legal marijuana
p hat + Z x √P(1P)/n
Event is the success part
P hat= 480/1004 = 6.478
C.I. legal= 0.478 + 1.96 x √0.478 (10.478)/1004 = 0.447, 0.510
95% C.I. 0.447, 0.510
We are 95% confident that the true proportion of PSU UP undergraduates who think marijuana
should be legalized is from 44.7% to 51.0%
90% confidence interval for same sex marriages
Minitab stat –basic stat one proportion select same sex put in correct C.I.
P hat = 216/1004= 0.215
0.215 + 1.65 √0.215 (1 0.215)/1004
90%/ 0.194, 0.236 We are 90% confident that the true proportion of PSU UP undergraduates that believe same sex
marriage should be permitted is from 19.4% to 23.6%
Mean (quantitative) Question
99% confidence for mean book cost
X bar + t x s/√n
Minitab stat basic stat sample t
X bar = 335.39 n= 997 S.D.= 137.21 D.F.= n1 9971= 996
Go to table A2 under 99% then use the one that is closest without exceeding it so now we use 100
which is 2.63
335.39 + 2.63 x 137.21/√997
99% C.I. 324.17, 346.60
We are 99% confident that the true mean amount of PSU UP undergraduates spend on books in a
semester is from $324.17% to $346.60%
1. Are these intervals confidence statements or probability statements? Confidence
2. What happens to margin of error as confidence increases? Margin of error
3. As confidence increases the interval gets wider. 10/19/10 Week 9
Null Hypothesis a statement that there is nothing happening. The specific null hypothesis varies
from problem to problem, but generally it can be thought of as the status quo, or no relationship, or
no difference. In most situations, the researcher hopes to disprove or reject the null hypothesis.
No extrasensory perception
No difference between the mean pulse rates of men and women
No relationship between exercise intensity and the resulting aerobic benefit.
Alternative Hypothesis a statement that something is happening. In most situations, this
hypothesis is what the researcher hopes to prove. It may be a statement that is assumed status
quo is false, or that there is a relationship, or that there is a difference.
Men have a lower pulse rate than women do
Increasing exercise intensity increases the resulting aerobic benefit.
Null Value the specific number the parameter equals if the null hypothesis is true.
Test Statistic computing the data summery that is used to evaluate the two hypothesis.
Pvalue is computed by assuming that the null hypothesis is true and then determining the
probability of a result as extreme (or more extreme) as the observed test statistic in the direction of
the alternative hypothesis.
Level of Significance the decision is made to accept the alternative hypothesis if the pvalue is
smaller than a designed level of significance, alpha, and usually set by researcher at .05, less
commonly at .1