• Today: Hypothesis testing
• Do not forget assignment 4… will increase your exam mark!!!!! • Four Basic Steps:
1. State hypothesis (in terms of pop. parameter)
2. Calculate test statistics
3. Find p-value
4. State Conclusion in terms of real problem at hand • Data: random sample x , x , 1, x2from a n(µ, σ) population
• Mean, µ, is unknown
• Standard deviation, σ, is known
• For testing the hypothesis H : µ=µ 0 0
x − µ0
• Test Statistic: z =
σ / n • Computing p_value depends on the alternate hypothesis:
Alternate hypothesis P-Value
H 1 µ > µ 0 P(Z ≥ z)
H 1 µ < µ 0 P(Z ≤ z)
H 1 µ ≠ µ 0 2P(Z ≥ z ) • A student group claims that first year students at a university
study 2.5 hours per weeknight
• A skeptical statistics professor claims that this is waaaay too
high
• A random sample of 269 university students found an average
study time of the students to be 137 minutes
• Suppose that the study times follow a normal distribution with
standard deviation of 65 minutes
• Using these data test the prof’s hypothesis with a sig. level of
0.01 • Hypotheses:
• Test Statistic
• P-value
• Conclusion • IQ test scores of 7 grade girls in the Midwest USA follow a
normal distribution with standard deviation of 15
• A random sample of 31 7 grade girls from a Midwest school
district is taken and their IQ’s measured…giving a sample mean
of 104
• IQ’s in the broad population supposed to have an average of
100
• Is there evidence at a 0.05 level that the mean in this district is
different from the mean in the general population? • Hypotheses:
• Test Statistic
• P-value
• Conclusion • For these data, would a 95% confidence interval contain the
hypothesized value ? • A level α 2-sided significance test rejects the hypothesis
H :µ=µ exactly when the value µ falls outside of the (1-α)100
0 0 0
percent confidence interval
• What does this mean? • Data must be a random sample
• Outliers can distort results
• Shape of the population distribution matters (large and small
samples?)
• Need to know σ
• For significance tests, there is a difference between practical
and statistical significance • Data must be a random sample
• Outliers can distort results
• Shape of the population distribution matters
• Need to know σ
• Margin of error does not cover all errors! • A survey of licensed drivers asked “Of every 10 motorists who
run a red light, about how many will be caught?”
• The mean result of 9 respondents was 1.92
• Suppose it is known from a previous survey that σ=1.83
• A histogram for the responses shows a right-skewed pattern
• A 95% confidence interval for the population mean is
desired….does it make sense to compute one with these data?
Why? • A survey of licensed drivers asked “Of every 10 motorists who
run a red light, about how many will be caught?”
• The mean result of 100 respondents was 1.92
• Suppose it is known from a previous survey that σ=1.83
• A histogram for the responses shows a right-skewed pattern
• A 95% confidence interval for the population mean is
desired….does it make sense to compute one with these data?
Why? • Suppose that the null hypothesis is true, we could collect a
sample that suggests that we reject H 0
• Suppose that H is 0ot true, we could fail to reject the null
hypothesis • To make inference about the population mean,Ж, we have
used the z-test
• Key feature: σ is known
• Mo

More
Less