Hypothesis Tests With
Means of Samples
Part 2: Sept. 9, 2014

• In estimating a population mean we have 2 options:
– 1) Point estimates give specific #
– Example:
– Accuracy of such a point estimate of the pop mean – ok, but not great
– Our sample may be unrepresentative, etc.

• 2) Interval estimates – provide range where you think pop mean may fall
– Ex) Given M=5.9 (which = μ
M)
σ
M)
, and std dev = .2 (which is
, and assuming a normal curve…
– We’d expect 34% of pop means to fall b/w 5.9 & 6.1
(+1 SD) and another 34% b/w 5.9 & 5.7 (-1 SD)
– 68% between 5.7 and 6.1  consider this a 68% confidence interval
• What does that mean?

• But 68% confident not that great…more interest in
95% or 99% confidence.
• Standard to use 95% or 99%
– Use normal curve table & find z score cutoffs for .05 significance level, 2-tailed test:
– Change these to raw scores for our example
• Using x = z( σ
M
) + M , what scores do you get?
• What is the final 95% confidence interval for this example and what does it mean?

• For 99% interval (2-tailed), find z score cutoffs in normal curve table:
• Change these to raw scores for our example
– x = z(σ
M
) + M, what do you get?
– What is the 99% confidence interval here? What does it mean?

• Notice the wider interval for 99% compared to narrower interval for 95%
– Wider  more likely you’re right and you include the actual mean in that interval

• Also note that we can calculate CIs for 1-tailed tests:
– You will still calculate 2 scores to give a range of confidence.
– What changes is the relevant z score…
– 95% CI for ‘crashed’ example:
• Z score cutoff will be 1.64, so use conversion formula:
• X = Z(σ
M
) + M and then x = -Z( σ
M
) + M , so…
• Resulting 1-tailed 95% CI is…
• Narrower or wider interval than 2-tailed 95% CI?

• As a shortcut, you may memorize or refer to these cutoff scores when computing CI’s – these will never change!
• Cutoff scores most often used:
– For a 95% CI, 1-tailed = 1.64 and –1.64
– 95% CI, 2-tailed = 1.96 and –1.96
– 99% CI, 1-tailed = 2.33 and –2.33
– 99% CI, 2-tailed = 2.57 and –2.57