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Never forget that even small effects can be statistically significant if the samples are large. To...

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 104 small businesses. During a three-year period, 10 of the 71 headed by men and 6 of the 33 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=0.05α=0.05)?
The P-value was so we conclude that
Choose a conclusion.The test showed strong evidence of a significant difference.The test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 180 out of 990 businesses headed by women and 300 out of 2130 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.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.The confidence interval's margin of error is reduced.The confidence interval's margin of error is increased.

Solutions

Expert Solution

a) The hypotheses are:

The sample proportions are:

The pooled proportion is:

The test statistic is:

The two tailed p-value is:

Since p-value is greater than , fail to reject the null hypothesis. The test showed no significant difference. We can conclude that the same proportion of women's and men's businesses fail.

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b) The hypotheses are:

The sample proportions are:

The pooled proportion is:

The test statistic is:

The two tailed p-value is:

Since p-value is less than , reject the null hypothesis. The test showed strong evidence of a significant difference. We cannot conclude that the same proportion of women's and men's businesses fail.

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c) The critical value of z at 95% confidence level is 1.96.

The 95% confidence interval for part a:

In this interval, margin of error is 0.154.

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The 95% confidence interval for part b:

In this interval, margin of error is 0.028.

The confidence interval's margin of error is reduced.


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