Statistics in a Modern World 800
Solutions to Practice Exam 4
- (a) Apparently fathers' mean age is higher than mothers' mean age,
because the interval for the difference of the means, (1.23, 4.40), clearly
contains only positive numbers.
- (c) Mean earnings could be the same, because the intervals (2,4) and
(3,5) share a substantial amount of overlap.
- (a) Sample mean weight may be biased; a shopper may tend to select
lighter, or heavier, berries.
-
- (i) z test about a proportion (the word "minority" implies the
test's alternative hypothesis is that the population proportion is less
than .5)
- (ii) z test about a mean (alternative hypothesis is that population
mean is less than 98.6)
- (iv) chi-squared test (for a relationship between two categorical
variables, gender and handedness)
- (b) very small P-value means very strong evidence of a relationship
- (ii) two-sided alternative makes it more difficult to reject the null
hypothesis, because the P-value is larger (twice the one-sided P-value)
-
- mean of sample proportions = population proportion = .5
- standard error = square root of (.5)(1-.5)/64 = .0625
- shape approximately normal because the sample is pretty large (64)
- 95% within 2 standard errors of the mean: between .5-2(.0625)=.375
and .5+2(.0625)=.625
-
- no: mean of sample proportions is always population proportion
(in this case, .5) no matter what the sample size
- yes: standard error goes from square root of (.5)(1-.5)/64 to
square root of (.5)(1-.5)/4
- yes: shape would be less normal because the sample size (4) is so
small
-
- (v) equal to .15 (proposed population proportion)
- (iv) less than .15 (we suspect population proportion is less than
what it is nationally)
- square root of (.15)(1-.15)/100 = .0357
- z = (.08-.15)/.0357 = -1.96
- Because the alternative hypothesis says "less than", P-value is
probability of z < -1.96 = .025
- yes: small P-value means reject null hypothesis and conclude the
alternative is true: population proportion is less than .15
- (iii) For a cut-off probability of .05, 5% of the time we will reject
the null hypothesis, even if it is true.
-
- 98.20 plus or minus 2(.6)/(square root of 100) = 98.20 plus or minus .12
= (98.08, 98.32)
- yes, because 98.6 is not in the interval
- (ii) more subjects yields a narrower interval because the margin of
error includes a DIVISION by square root of sample size
- (i) is the only correct interpretation. Rule out (ii) and (iii)
because the confidence interval is about MEAN temperature, not the
temperatures themselves. Rule out (iv) because population mean does NOT
vary; it is a fixed parameter. Rule out (v) because the confidence
interval is about POPULATION mean, not SAMPLE mean. In fact, we are
100% confident that sample mean is in the interval---we built our interval
around it!
- (a) a very small sample tends to produce a small z, which results in
a large P-value, and failure to reject
- (b) a very large sample tends to produce a large z, which results in
a small P-value, and strong evidence of a difference which may be unimportant
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