Statistical Reasoning
Solutions to Midterm 2
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- (iii) (large enough n ensures Central Limit Theorem applies)
- (ii) (pop greater than 10n ensures dependence doesn't undermine
formula for sd)
- PROPORTION has standard deviation = square root of (0.75(1-0.75))/115 = 0.04
- z=(0.80-0.75)/0.04=1.25
- (iii) because z is positive but not extreme
-
- (iii) 0.05(100)=5
- (viii) 0.95(100)=95
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- Use the 68-95-99.7 Rule: The answer is 16% because 5.5 is 1 standard deviation above the mean
- 5-1.82(0.5)=4.09
- Use the z curve shown: The top 1% have z=2.326 so you take 5+2.326(0.5)=6.163
-
- (i) The null hypothesis takes the point of view that
nothing is going on; in this case, that Peppa Pig videos do not cause autism. A Type I Error erroneously rejects a true Ho. (A lot of students missed this question. There will be a 5-point Extra Credit question on the Final Exam that tests your understanding of what constitutes Type I or Type II Errors.)
- (i) The null hypothesis always takes the point of view that
nothing is going on; in this case, that a drug doesn't lead to harmful
side effects. A Type II Error erroneously fails to reject a false Ho, and this can happen if the sample size is too small to catch occasional occurrences of side effects.
-
- (iii) Since t=-0.91 is not at all large in absolute value, the P-value
isn't small at all.
- (iv) because 377625>272500 and the P-value isn't small.
- (i) because the test tells us 0 is a plausible difference between
population means.
- (iii) because the StDev is almost as large as the Mean and prices can't be negative, so much more of the spread occurs to the right of the mean, as opposed to the left
-
- mean of sample means is population mean 1.4 and s.d. of sample means is
population s.d. over square root of sample size, or 1.2/2=0.6
- (iii) As we take larger samples, the shape becomes closer to normal.
However, the sample is quite small and the distribution is very right-skewed,
so sample mean will still have some right skewness.
-
- (iii) (it's the farthest ABOVE 5)
- (i) (it's the farthest AWAY FROM 5)
- 2(0.012)=0.024
-
- 0.51 plus or minus twice square root of 0.51(1-0.51)/400= (0.46, 0.56)
- (ii) because the multiplier would be less than 2 [not (i): smaller n
tends to give wider intervals]
- (ii) because the interval clearly contains 0.50
- (i) [It's making a claim about p, not p-hat, but p does not obey the
laws of probability, so we use the word "confidence".]
- (iii) as discussed on practice midterm
-
- (iii) (because standardized sample mean does not follow t distribution
when sample is small and non-normal)
- (i) (refer to t because n is small and sigma is unknown)
- (ii) (refer to z because n is large; an outlier isn't a problem if n is large)
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