Stats test 3
March 17 , 2014
Sec. 7.5 Determining the sample size needed to estimate μ or p
1. What n is needed to estimate p using a confidence interval (CI)?
p±Z α ▯Sampleing error = Zα
2 √ n 2√ n
n= 2 = Given: a) confidence level b) SE
A sociologist wants to estimate the % of US households using email. How many households
must be sampled to get a SE of no more than 4% and 95% confidence.
SE=.04 Z=1.96 p=? n= SE 2 1α=.95
If no info is given, use .5 for p
N= 1.96 ∗(25)(.=600.25=601
Any decimals in problems like this gets rounded up.
Exampl2: Suppose a295% CI, for p is (.20,.30). What was the value of n used?
Z pq 1.96 ∗.25∗.75
n= SE 2 = .52 = [.5=(.3.2)/2 and .25 is the middle of the CI]
2. What n is needed to estimate μ usng a confidence interval?
± Z∗σ →SE= Z∗σ
X(bar) √n √n
n= Z σ2 Note: σ2 or range of x’s is given. Range ≃ 4 σ
Worksheet #5: An estimate is needed of the average age of customers in a department store
correct to within 2 years with prob .98. How many sustomers should be smapled? Assume age
range =15 to 45.
1 ∉α=.98SE=2range=30Z=2.33σ= 40=7.5
n= 2.33 (7.5)=76.34=77
Questions to ask:
1.What is the parameter? μ or p
2. what is the sample size? Stats test 3
To estimate the average monthly rent for 1BDR apartment, 12 complexes are randomly
selected. Mean cost =$980 s=87
X(bars) ±t√n df=n1=121=11
980 ± (934.89,1025.11)
How many patients need to be on a diet plan to estimate weight loss to within 2 months
n ¿2.32 6=4.9
Suppose the sample is taken from #7 and the average weight loss 12.0 lbs. Find 98% CI for the
true mean weight loss.
12.6 ± = (10.6,14.6)
Find n 96% CI .09=width
N= .09/2=SE n=475
A sample is taken and 60% support. Find 95% CI. (This is from #9)
.6 ±1.96 = .6.044
CH.8 Test of Hypothesis Stats test 3
Elements of a test of hypothesis (sec 8.1)
Introduction: A claim is made about a parameter ( μ∨p¿. Take a sample and decide if you believe it or not.
Element of test: H0, Ha, Test statistic, rejection, region, conclusion.
1. Null hypothesis or H
a. A statement of equality about the parameter
i. H 0 μ−50
ii. H 0 p=.3
2. Alternative Hypothesis or H (Aaso known as the research hypothesis)
a. A statement of inequality ( ≠,.3
3. Test Statistic
a. Calculate a number that compares the sample info to what is hypothesized.
b. Example: μ,n≥30
c. Z= σ that’s an x bar**
Suppose it is claimed that average age of UCF students is 27. To check the claim a sample of 100 students is
taken and their average age and standard deviation is calculated.
Is there evidence to suggest the average age is actually less than 27?
H 0 μ=27 a : μ<27
1. n=100 s=5 x(bar)=26.5
a. z= =−1
b. can’t say μ<27 Do not reject H0. ere is not evidence that H ais true.
2. n=100 s=5 x(bar)=25.5
a. z= 25.5−27 =−3
i. Yes, I think μ<27 Reject H . There is evidence that H is true.
3. N=100 s=5 x(bar)= 28.5
a. z= 5/ 100 =3
b. can’t say μ <27. Do not reject H 0. re is not evidence that H as true. hen you
find that x=bar is larger then the μthenthat shouldbe acluethat youcan t sothat.
4. Rejection region (RR)
a. Gives unusual values of the test statistic if 0 is true.
b. Z=2 is the cut off point in the tail of a bell curve. Z=0 is in the middle
c. Rejection region is the left side of the 2 cut off point. (left tail)
d. If your statistic falls within the rejection region you can reject0H . (RR: z (upper tail test)
a Stats test 3
a. Ex) a=.05 Z a 1.645
3. H ais ▯two tail test
a. Ex) a=.025 RR: Z1.96
If the test statistic fail in the RR, you reject o
Reject H 0▯evidence to conclud