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Lecture

Lecture Set 5

32 Pages
115 Views

Department
Statistics
Course Code
STAT 203
Professor
Derek Bingham

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Description
•  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
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