Randomization is the most important part of testing'

1
Test 5 Significance Testing

• Randomization is the most important part of
  testing.
• When the sample mean, x, is unbiased it is
  approximately equal to the population mean, m.
• Unbiased means totally random.

2
Confidence Intervals

• Remember that the sample mean, x, is an estimate
  of the parameter, m.
• You can give an interval of numbers that you are
  confident that the real mean (parameter) is
  between.
• The formula is

• z is based on C, the Confidence Level (bottom of
  Table C)

3
Margin of Error and Sample Size

• Margin of error is calculated using the following
  formula

• Margin of error explains error due to variation
  not due to sampling errors.

• You can decide the sample size needed to obtain a
  particular confidence level and margin of error.
  The formula is

4
Assumptions

• The distribution is approximately normal
• You must know s - which is unrealistic (we will
  learn in chapter 6 how to determine this)

• SRS
• If x is influenced by outliers, it may be
  necessary to remove them.
• n is fairly large (ngt15)

5

• A New York Times poll on womens issues
  interviewed 1025 women randomly selected from the
  US, excluding Alaska and Hawaii. The poll found
  that 47 of the women said they do not get enough
  time for themselves.
• (a) The poll announced a margin of error of
  plus or minus 3 percentage points for 95
  confidence in its conclusions. What is the 95
  confidence interval for the percent of all adult
  women who think they do not get enough time for
  themselves?
• (b) Explain to someone who knows no statistics
  why we cant just say that 47 of all adult women
  do not get enough time for themselves.
• (c) Then explain clearly what 95 confidence
  means.

6
(a) 44 - 50
(b) small sample of a large population not exact
(c) Were only 95 sure that between 44 - 50
of women dont get enough time for themselves
7
The Degree of Reading Power (DRP) is a test of
the reading ability of children. Here are DRP
scores for a sample of 44 third-grade students in
a suburban school district.
8
Answer the following

• We expect the distribution of DRP scores to be
  close to normal. Make a stemplot or histogram of
  the distribution of these 44 scores and describe
  its shape.
• Suppose that the standard deviation of the
  population of DRP scores is known to be 11. Give
  a 99 confidence interval for the mean score in
  the school district.
• Would you trust your conclusion from (b) if these
  scores came from a single class in one school in
  the district? Why?

9
Example 2
(a) approximately normal
(b)
Notice that as Confidence goes up, the margin of
error increases. As Confidence goes down, the
margin of error decreases.
(c) no undercoverage
10
The National Assessment of Educational Progress
test was given to a sample of 1077 women (ages 21
- 25). Their mean quantitative score was 275 and
the standard deviation is 60.

• Give a 95 confidence interval for the mean
  score.
• Give the 90 and 99 confidence intervals.
• What are the margins of error for 90, 95, and
  99 confidence? How does increasing the
  confidence level affect the margin of error of a
  confidence interval?

11
Example 3
(a) a little skewed right but approximately
normal
(b)
12
The NAEP sample of 1077 young women had mean
quantitative score 275 and standard deviation 60.

• Give a 95 confidence interval for the mean
  score.
• Suppose that the same result of 275 came from a
  sample of 250 women. Give a 95 confidence
  interval for this case.
• Suppose that 4000 women were tested -- give a 95
  confidence interval for this case.
• What are the margins of error for size 250, 1077,
  and 4000? How does changing the sample size
  change the margin of error?

13
Example 4
(a) C 95
(b) C 90 C 99
(c) As C goes up, m increases and as C goes
down, m decreases
14
To assess the accuracy of a laboratory scale, a
standard weight known to weigh 10 grams is
weighed repeatedly. The scale readings are
normally distributed with unknown mean (if the
scale has no bias). The standard deviation is
0.0002.

• The weight is weighed five times. The mean
  result is 10.0023 grams. Give a 98 confidence
  interval.
• How many measurements must be averaged to get a
  margin of error of plus or minus .0001 with 98
  confidence?

15
Example 5
(a) C 98 (10.0021, 10.0025)
(b) m .0001 z2.326