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1-5: Constructing Confidence Intervals
1-5
Vocabulary:
Point Estimate- the value of a statistic used to estimate the parameter.
(this is because it consists of a single number or point)
Confidence Interval (CI) – An interval of numbers obtained from a point estimate of a parameter.
Confidence Level (CL) – The confidence we have that the parameter lies in the confidence interval (basically the
level that the confidence interval WILL contain that parameter).
Confidence-Interval Estimate – Ranges of values used to estimate some population parameter with a specific
confidence level.
Central Limit Theorem – For a relatively large sample size, the variable x is approximately normally distributed,
regardless of the distribution of the variable under consideration. The approximation becomes better with
increasing sample size.
Thus:
µ = µx which means the population mean is equal to the average of all of the sample means.
σx = σ / √n
Which means the standard deviation of all of the sample standard deviations is equal to the population
standard deviation divided by the square root of the sample size (n).
Margin of Error – The difference between the population parameter and the observed sample statistic.
Z-score – Number of standard deviations that a value is above or below a mean.
Population:
Sample:
Population Standard Deviation
- The mean of a small population is easy to compute. However, the mean for a large population can be impractical
or even impossible.
We have previously discussed taking a sample from a large population to gain a reasonable estimate of information
for the population.
Examples:
1. 21 students were surveyed about their drive time to Lakes. Each student’s response is listed below.
1.5, 3, 3.5, 3.5, 4, 4.5, 5, 5, 6, 6.5, 6.5, 6.5, 6.5, 7.5, 8.5, 9, 9.5, 9.5, 10, 10, 13.5
Find the z-score for each value, with respect to the data above.
a. 12.4
b. 3.7
c. 10
d. 16
2. How confident are you???
Sample Size ( n ) =
Sample Mean ( 𝑥̅ ) =
Sample Standard Deviation ( s ) =
Sampling Error ( se ) =
Construct a confidence interval at 68%, 95%, and 99.7%.
What happens if we change the sample size?
Construct new confidence intervals at 68%, 95%, and 99.7% with a sample size of _______.
e. 0.5
3. Let’s get some samples for the rectangles!!!
Sampling Distribution Practice, use a sample size of 10 rectangles
Write down the sample means for each sample here:
Construct a Confidence Interval for 90%, 95% and 99%. The z-scores are 1.645, 1.96, and 2.58 respectively.
CAUTION!!!
Misconception 1: A 95% confidence interval does NOT mean that for an interval calculated from sample data there
is a 95% probability the population parameter lies within the interval.
Misconception 2: A 95% confidence interval does NOT mean that there is a 95% probability that the interval
covers the population parameter.
Misconception 3: A 95% confidence interval does NOT mean that 95% of the sample data lie within the interval.
Misconception 4: A confidence interval is NOT a range of plausible values for the sample mean, though it may be
understood as an estimate of plausible values for the population parameter.
Misconception 4: A particular confidence interval of 95% does NOT mean that there is a 95% probability of a
sample mean from a repeat of the experiment falling within this interval