Basic Applied Statistics 200
Solutions to MIDTERM 2

  1.  
    1. 10.4-2(4.6)=1.2 to 10.4+2(4.6)=19.6
    2. P(X>20)=P(Z>(20-10.4)/4.6)=P(Z>+2.09)=P(Z<-2.09)=.0183
  2.  
    1. mean of sample means is population mean 100, s.d. of sample means is population s.d. over square root of sample size, or 10.61
    2. (iv) shape of distribution of sample mean is still right-skewed because the sample size (8) is so small
  3.  
    1. (i) population should be at least 10 times sample size
    2. (ii) sample must be large enough to use a normal approximation
    3. COUNT X has mean np=100(.8)=80 and sd square root of np(1-p) = 4.0
    4. PROPORTION has mean p=.8 and sd square root of p(1-p)/n = .04
    5. P(sample prop.<.70)=P(Z<(.7-.8)/.04)=P(Z<-2.5)=.0062
    6. (ii) because of the small probability in (e)
    7. (i) is the same as rejecting the null hypothesis when it's true
    8. (i) since Ho was rejected in (f)
    9. (ii) a statistic called p-hat
  4.  
    1. (viii) In the long run, 95% of the 200 intervals, or 190, should contain p.
    2. (iv) In the long run, 5% of the 200 tests, or 10, should reject a true Ho.
  5.  
    1. .38 plus or minus .03=(.35,.41)
    2. (ii) narrower
    3. (i) wider
    4. (i) yes because .40 is in the interval
    5. (iii) both are biased: (i) because of a non-representative sample and (ii) because of a preceding question
    6. (iii) is the correct interpretation
  6.  
    1. (vi) is the correct interpretation
    2. Ho: mu=1 vs. Ha: mu not equal to 1
    3. t=(1.3-1)/(.5/square root of 15)=2.32
    4. p-value between 2(.025) and 2(.01): in (.02, .05)
    5. (i) Since the p-value is small, we reject Ho and conclude there is convincing evidence that the alternative is true: population mean number of siblings is not 1.
    6. .01 to .025
  7.  
    1. (i) yes: two scores recorded as a pair for each student
    2. n minus 1 = 17-1=16
    3. (iii) can't quite reject Ho, because the p-value is just over .05
    4. (i) OK because the histogram is roughly bell-shaped
    5. (ii) the CI just barely includes zero
    6. (iii) which quiz is taken constitutes the categorical explanatory variable


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