Download + 1.96

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Degrees of freedom (statistics) wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Resampling (statistics) wikipedia, lookup

German tank problem wikipedia, lookup

Misuse of statistics wikipedia, lookup

Student's t-test wikipedia, lookup

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’.
__________________________________________________________________
Recall that sample means are normally distributed if the population is normal or if
n ≥ 30.
__________________________________________________________________
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 ).
__________________________________________________________________
However, if σ is not known, we will use t-scores:
𝑡=
𝑥− 𝜇
𝑠
√𝑛
__________________________________________________________________
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
__________________________________________________________________
𝑡𝛼 ∙
2
𝑠
√𝑛
is called the ‘margin of error’ (half the length of the confidence interval)
__________________________________________________________________
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?
__________________________________________________________________
Use Stats/Tests/TInterval
__________________________________________________________________
Example: (p.450, #36ac)
[Note text answer.]
__________________________________________________________________
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?
__________________________________________________________________
Determining Sample Size
𝑧𝛼⁄2 ∙𝑠 2
𝑛=(
𝐸
)
[round up]