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"Kind words can be short and easy to speak, but
their echoes are truly endless - Mother Teresa
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Two Sample Means Problem
The board of directors at the Anchor Pointe
Marina is studying the usage of boats among its
members. A sample of 30 members who have boats
10 to 20 feet in length showed that they used
their boats an average of 11 days last July. The
standard deviation of the sample was 3.88 days.
For a sample of 40 member with boats 21 to 40
feet in length, the average number of days they
used their boats in July was 7.67 with a standard
deviation of 4.42 days. At the .02 significance
level, can the board of directors conclude that
those with the smaller boats use their crafts
more frequently?
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Step 1
State the null and alternative hypothesis. H0
Large boat usage small boat usage H1 Smaller
boat usage gt large boat usage
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Step 2
Select a level of significance. This will be
given to you. In this problem it is .02.
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Step 3
Formulate a decision rule. .5000 - .0200 .4800
2.05z
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Step 4
Identify the test statistic.
11 7.67 3.35z 3.882 4.422
30 40
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Step 5
Arrive at a decision. The test statistic
falls in the critical region, therefore we reject
the null.
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p-Value in Hypothesis Testing

• p-Value The probability, assuming that the null
  hypothesis is true, of getting a value of the
  test statistic at least as extreme as the
  computed value for the test.
• If the p-value area is smaller than the
  significance level, H0 is rejected.
• If the p-value area is larger than the
  significance level, H0 is not rejected.

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Statistical Significance
p-Value The probability of getting a sample
outcome as far from what we would expect to get
if the null hypothesis is true. The stronger
that p-value, the stronger the evidence that the
null hypothesis is false.
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Statistical Significance
P-values can be determined by - computing the
z-score - using the standard normal table The
null hypothesis can be rejected if the p-value is
small enough.
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P-Value
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Tests Concerning Proportions

• Proportion
• A fraction or percentage that indicates the part
  of the population or sample having a particular
  trait of interest.

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Tests Concerning Proportions

• The sample proportion is denoted by p, where
• p number of successes in the sample
• number sampled

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Test for One Proportion
p population proportion p sample
proportion
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Break Time
Break Time
Party, Party, Party!!!! Statistics is almost
over.
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One Sample Proportion Problem
An urban planner claims that, nationally, 20
percent of all families renting condos move
during a given year. A random sample of 200
families renting condos in Dallas revealed that
56 had moved during the past year. At the .01
significance level, does this suggest that a
larger proportion of condo owners moved in the
Dallas area? Determine the p-value.
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Step 1
State the null and alternative hypothesis. H0
Proportion .20 H1 Proportion gt .20
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Step 2
Select a level of significance. This will be
given to you. In this problem, it is .01.
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Step 3
Formulate a decision rule. .5000 - .01 .4900
2.32z
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Step 4
Identify the test statistic. Z
.28 - .20 2.83z .20(1-.20)
200
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Step 5
Arrive at a decision. The test statistic
falls in the critical region, therefore, we
reject the null.
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Test for Two Proportions
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Two Proportion Problem
Suppose that a random sample of 1,000
American-born citizens revealed that 198 favored
resumption of full diplomatic relations with
Cuba. Similarly, 117 of a sample of 500
foreign-born citizens favored it. At the .05
significance level, is there a difference in the
proportion of American-born versus foreign-born
citizens who favor restoring diplomatic relations
with Cuba?
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Step 1
State the null and alternative hypothesis. H0
Proportion of American-born Foreign born H1
Proportion of American-born ? Foreign-born
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Step 2
Select a level of significance. This will be
given to you. In this problem it is .05.
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Step 3
Formulate a decision rule. 1.000 - .0500
.9500 .9500/2 .4750 1.96z
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Step 4 Part I
Identify the test statistic. PC
198 117 .21 1000 500
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Step 4 Part II
Identify the test statistic. Z
.198 - .234
-1.61z .21(1-.21) .21(1-.21) 1000
500
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Step 5
Arrive at a decision. The test statistic
falls in the null hypothesis region, therefore we
fail to reject the null.
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Type I and Type II Errors

• Type I Error
• Rejecting the null hypothesis when H0 is actually
  true.
• Type II Error
• Accepting the null hypothesis when H0 is actually
  false.

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Type I Error
Rejecting the null hypothesis when H0 is
actually true.
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Type II Error
Accepting the null hypothesis when H0 is
actually false.