Introduction to inference
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Introduction to Inference. Confidence Intervals for Proportions. Example problem. In a study of air-bag effectiveness, it was found that in 821 crashes of midsize cars equipped with air bags, 46 of the crashes resulted in hospitalization of the drivers.

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Introduction to Inference

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Introduction to Inference

Confidence Intervals for Proportions


Example problem

  • In a study of air-bag effectiveness, it was found that in 821 crashes of midsize cars equipped with air bags, 46 of the crashes resulted in hospitalization of the drivers.

    • Give a 95% confidence interval for the percent of crashes resulting in hospitalization. Interpret the confidence interval.


Sample means to sample proportions

parameter

statistic

mean

proportion

standard deviation

Formulas:


Confidence Intervals for proportions

Draw a random sample of size n from a large

population with unknown proportion p of successes.

Formula:

Z-interval

One-proportion Z-interval


Conditions for proportions

  • The data are a random sample from the population of interest.

  • Issue of normality:

    • np > 10 and n(1 – p) > 10

  • The population is at least 10 times as large as the sample.


In a study of air-bag effectiveness, it was found that in 821 crashes of midsize cars equipped with air bags, 46 of the crashes resulted in hospitalization of the drivers.

Give a 95% confidence interval for the percent of crashes

resulting in hospitalization.

1 proportion z-interval

We assume the sample is a random sample.

Sample size is large enough

to use a normal distribution.

Safe to infer population is at least 8210 crashes.


Give a 95% confidence interval for the percent of crashes resulting in hospitalization.

We are 95% confident that the true proportion of crashes

lies between .0403 and .0718.

Since we had to assume the crashes were a random sample,

we have doubts about the accuracy.


How large a sample would be needed to obtain the same margin of error in part “a” for a 99% confidence interval?

We need a sample size

of at least 1419 crashes.


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