Confidence level and critical values

About this lesson

Statistics (AP-aligned) › Confidence intervals

To save a PDF, choose “Save as PDF” in the print dialog.

Confidence level and critical values

Warm-up

Three quick questions from earlier lessons.

Discover

To be more confident of catching a fish, should you use a wider or a narrower net?

  1. Wider
  2. Narrower

Key idea

The confidence level describes the method.

Common critical values: z∗=1.645z^* = 1.645 (90%), 1.961.96 (95%), 2.5762.576 (99%). Higher confidence gives a wider interval. The confidence level is the proportion of intervals, in repeated random sampling, that capture the parameter.

Worked example

p^=0.6\hat{p} = 0.6, SE=0.04SE = 0.04.

90%: ME ≈ 0.066
1.645 × 0.04.
95%: ME ≈ 0.078
1.96 × 0.04.
99%: ME ≈ 0.103
2.576 × 0.04.

Your turn, guided

Your turn: SE = 0.05, 95% (z* = 2 approx). The margin of error.

0.10.1
2 × 0.05.

Practice ★

  1. From the same sample of 458, a 99% and a 90% interval are computed. Which is wider?
    1. The 99% interval
    2. The 90% interval
    3. They have the same width
  1. A 95% interval (z* = 1.96) has margin of error 0.07. From the same data, find the margin of error of a 99% interval (z* = 2.576).
  2. A 95% interval (z* = 1.96) has margin of error 0.06. From the same data, find the margin of error of a 99% interval (z* = 2.576).

Watch out!

Practice ★★

  1. What does the confidence level 90% mean?
    1. In repeated random sampling, about 90% of the intervals built this way capture the parameter
    2. This interval contains the parameter with probability 90%
    3. 90% of the data fall in the interval

Exit ticket

Three questions to check you have got it.

  1. A 95% interval (z* = 1.96) has margin of error 0.07. From the same data, find the margin of error of a 99% interval (z* = 2.576).
  2. A 95% interval (z* = 1.96) has margin of error 0.06. From the same data, find the margin of error of a 99% interval (z* = 2.576).
  3. A 95% interval (z* = 1.96) has margin of error 0.09. From the same data, find the margin of error of a 99% interval (z* = 2.576).

Summary card

Confidence level

z∗z^*: 1.645, 1.96, 2.576. Higher confidence, wider interval.

Watch out: the level describes the method over many samples.

End of the lesson

Next lesson: Choosing a sample size Back to the chapter

Answers: Confidence level and critical values

A

  1. The 99% interval

B

  1. 23250\frac{23}{250}
  2. 69875\frac{69}{875}

C

  1. In repeated random sampling, about 90% of the intervals built this way capture the parameter

D

  1. 23250\frac{23}{250}
  2. 69875\frac{69}{875}
  3. 2071750\frac{207}{1750}

CC BY-SA 4.0