In: Statistics and Probability
Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 90 small businesses. During a three-year period, 10 of the 73 headed by men and 3 of the 17 headed by women failed.
(a) Find the proportions of failures for businesses headed by
women and businesses headed by men. These sample proportions are
quite close to each other. Give the P-value for the test of the
hypothesis that the same proportion of women's and men's businesses
fail. (Use the two-sided alternative). What can we conclude (Use
α=0.05)?
The P-value was
so we conclude that
The test showed strong evidence of a significant difference or The test showed no significant difference.
(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 90 out of 510 businesses headed by women and 300 out of 2190 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part
(a). Repeat the test for the new data. What can we
conclude?
The P-value was
so we conclude that
Choose a conclusion. The test showed strong evidence of a
significant difference or The test showed no significant
difference.
(c) It is wise to use a confidence interval to estimate the size
of an effect rather than just giving a P-value. Give 95% confidence
intervals for the difference between proportions of men's and
women's businesses (men minus women) that fail for the settings of
both (a) and (b). (Be sure to check that the conditions are met. If
the conditions aren't met for one of the intervals, use the same
type of interval for both)
Interval for smaller samples: __to__
Interval for larger samples: __to__
What is the effect of larger samples on the confidence
interval?
Choose an effect. The confidence interval is unchanged or The
confidence interval's margin of error is reduced or The confidence
interval's margin of error is increased.
(a)
Hypotheses are:
The proportions of failures for businesses headed by women is
The proportions of failures for businesses headed by men is
Pooled sample proportion is
Standard error of the test is:
So test statistics will be
The p-value is:
p-value = 2P(z > 0.42) = 0.6744
Since p-value is greater than α=0.05 so we fail to reject the null hypothesis.
The test showed no significant difference.
(B)
Hypotheses are:
The proportions of failures for businesses headed by women is
The proportions of failures for businesses headed by men is
Pooled sample proportion is
Standard error of the test is:
So test statistics will be
The p-value is:
p-value = 2P(z > 2.28) = 0.0226
Since p-value is lesser than α=0.05 so we reject the null hypothesis.
The test showed strong evidence of a significant difference.
(c)
For part a:
Let us find the standard error of estimate so
For 95% confidence interval z-value will be 1.96 so required confidence interval is
Hence, the required confidence interval is (-0.1581, 0.2371).
For part b:
Let us find the standard error of estimate so
For 95% confidence interval z-value will be 1.96 so required confidence interval is
Hence, the required confidence interval is (0.0034, 0.0756).
The confidence interval's margin of error is reduced.