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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

Introduction to Inference

Confidence Intervals for Proportions


Example problem
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
Sample means to sample proportions

parameter

statistic

mean

proportion

standard deviation

Formulas:


Confidence intervals for proportions
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
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
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 resulting in hospitalization.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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