Hypothesis Testing For a Single Population Mean

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Hypothesis Testing For a Single Population Mean
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Example Grade inflation?
Population of 5 million college students
Is the average GPA 2.7?
How likely is it that 100 students would have an
average GPA as large as 2.9 if the population
average was 2.7?
Sample of 100 college students
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The p-value illustrated
How likely is it that 100 students would have an
average GPA as large as 2.9 if the population
average was 2.7?
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Determining the p-value
H0 µ average population GPA 2.7 HA µ
average population GPA gt 2.7
If 100 students have average GPA of 2.9 with
standard deviation of 0.6, the P-value is
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Making the decision

• The p-value is small. It is unlikely that we
  would get a sample as large as 2.9 if the average
  GPA of the population was 2.7.
• Reject H0. There is sufficient evidence to
  conclude that the average GPA is greater than
  2.7.

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Terminology

• H0 µ 2.7 versus HA µ gt 2.7 is called a
  right-tailed or a one-sided hypothesis test,
  since the p-value is in the right tail.
• Z 3.33 is called the test statistic.
• If we think our p-value is small if it is less
  than 0.05, then the probability that we make a
  Type I error is 0.05. This is called the
  significance level of the test. We say,
  a0.05, where a is alpha.

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Example Body Temperature?
Population of many, many adults
Is average adult body temperature 98.6 degrees?
Or is it lower?
Average body temperature of 80 sampled adults is
98.4 degrees.
Sample of 80 adults
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The p-value illustrated
How likely is it that 80 adults would have an
average body temp as small as 98.4 if the popn
average was 98.6?
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Determining the p-value
H0 µ average popn body temp 98.6 HA µ
average popn body temp lt 98.6
If 80 adults have average body temp of 98.4 with
standard deviation of 0.6, the P-value is
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Making the decision

• The p-value is small. It is unlikely that we
  would get a sample as small as 98.4 if the
  average body temp of the population was 98.6.
• Reject H0. There is sufficient evidence to
  conclude that the average body temp is smaller
  than 98.6.

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Terminology

• H0 µ 98.6 versus HA µ lt 98.6 is called a
  left-tailed or a one-sided hypothesis test,
  since the p-value is in the left tail.
• Z -2.98 is the test statistic.
• If we think our p-value is small if it is less
  than 0.02, then the probability that we make a
  Type I error is 0.02. That is, significance
  level a 0.02.

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Example on Alcohol?
Population of Penn State students
Is average amount spent weekly 20?
Average amount spent is 17 with standard
deviation of 16.
Sample of 64 students
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The p-value illustrated
How likely is it that 64 students would spend an
average as small as 17, or as large as 23, if
the popn avg was 20?
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Determining the p-value
H0 µ average spent 20 HA µ average
spent ? 20
If 64 students spend an average of 17 with
standard deviation of 16, the P-value is
and

So P-value 0.067 ? 2 0.134
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Making the decision

• The p-value is not small. It is likely that we
  would get a sample as small as 17, or as large
  as 23, if the average amount spent on alcohol
  was 20.
• Do not reject H0. There is not enough evidence
  to conclude that the average amount spent differs
  from 20.

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Terminology

• H0 µ 20 versus HA µ ? 20 is called a
  two-tailed or a two-sided hypothesis test,
  since the p-value is in both tails.
• Z -1.5 is the test statistic.
• Since we failed to reject the null hypothesis, we
  may have made a Type II error.

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Using Minitab for HT for Mean

• If data are normally distributed
• Select Stat, Basic Statistics, and 1-sample t
• Select variable.
• Select Test mean. In box, specify the value of
  the mean in the null hypothesis.
• Select appropriate alternative hypothesis.
• Select OK. Output appears in session window.
• If data arent normally distributed, must have a
  large sample.

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Very Important Point

• Your p-value will not be correct unless the
  assumptions are correct!!!!
• If you have a small sample, check to see if your
  data are normally distributed!!
• If data are not normally distributed, you must
  have a large sample!