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Transcript
9.2
Confidence Intervals for Population Means (σ unknown)
Confidence intervals: We wish to use the sample mean as a point estimate of the
population mean. The best we can do is construct an interval which contains the
mean, with a certain ‘level of confidence’.
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Recall that sample means are normally distributed if the population is normal or if
n ≥ 30.
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Suppose we know σ. We can calculate 𝜎𝑥̅ .
We also know 95% of all sample means lie within 1.96 s.d. of μ. [use
invNorm(.975) ≈ 1.96]
i.e. –
μ – 1.96𝜎𝑥̅ < 𝑥̅ < μ + 1.96𝜎𝑥̅ (95% chance of being true)
or
𝑥̅ – 1.96𝜎𝑥̅ < 𝜇 < 𝑥̅ + 1.96𝜎𝑥̅
which gives us a “95% confidence interval” of (𝑥̅ – 1.96𝜎𝑥̅ , 𝑥̅ + 1.96𝜎𝑥̅ ).
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However, if σ is not known, we will use t-scores:
𝑡=
𝑥− 𝜇
𝑠
√𝑛
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These scores are not normally distributed. Their distribution is called the
Student’s t-distribution (with n-1 degrees of freedom).
Properties of the t-distribution
1. Distribution varies as n varies.
2. Bell-shaped and symmetric about 0.
3. Area under the curve is 1.
4. Graph approaches zero in both tails.
5. Area in tails is slightly larger than for normal distribution.
A (1-α)100% confidence interval will be given by:
lower bound: 𝑥̅ − 𝑡𝛼 ∙
2
𝑠
√𝑛
upper bound: 𝑥̅ + 𝑡𝛼 ∙
2
𝑠
√𝑛
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𝑡𝛼 ∙
2
𝑠
√𝑛
is called the ‘margin of error’ (half the length of the confidence interval)
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Example: (3rd ed.)
A random sample of size n is drawn from a population that is normally distributed.
The sample mean is found to be 108 and the sample standard deviation is found to
be 10.
a. Construct a 96% confidence interval for μ if the sample size is 25.
b. Construct a 96% confidence interval for μ if the sample size is 10.
c. Construct a 90% confidence interval for μ if the sample size is 25.
Compare the above answers, noting relationships of n, α and the margin of error.
d. Could we have computed the above confidence intervals if the population was
not normally distributed?
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Use Stats/Tests/TInterval
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Example: (p.450, #36ac)
__________________________________________________________________
Practice: (3rd ed.)
A random sample has a mean of 18.4 and a standard deviation of 4.5.
a. Construct a 95% confidence interval for μ if the sample size is 35.
b. Construct a 95% confidence interval for μ if the sample size is 50.
c. Construct a 99% confidence interval for μ if the sample size is 35.
d. If the sample size was 15, what conditions must be satisfied to compute the
confidence interval?
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Determining Sample Size
𝑧𝛼⁄2 ∙𝑠 2
𝑛=(
𝐸
)
[round up]
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