15D - Confidence intervals
Crash course statistics: Confidence intervals
This video covers the theory and some example of confidence intervals and their interpretations.
■ Warning! The content in the video after 7:00 minutes is not relevant to the VCE Mathematical Methods Study Design. |
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■ When the population proportion, \(p\), is unknown we can estimate its value using a sample proportion, \(\hat{p}\). However, the point estimate is unlikely to be the actual population parameter. Hence, we use \(\hat{p}\) to create an interval of numbers where the population parameter is likely to exist. This is known as a confidence interval.
- \(\hat{p}\) is a point estimate for the population proportion. It is unlikely that the sample proportion is equal to the population proportion; however, it should be close to it.
- The confidence interval does not have to contain the true population proportion, but it is likely to.
The 95% confidence interval:
Using knowledge of the standard normal distribution, we know that: \[Pr(-1.96<Z<1.96)=0.95\] Using this information, and our knowldge of the distribution of population proportions, \(\hat{P}\), we can construct an approximate 95% confidence interval for the unknown population proportion, \(p\): \[\left ( \hat{p}-1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p}+1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right )\]
15D - Example 1: Calculating an approximate 95% confidence interval
In Australia, 200 random people were surveyed and it is found the 34 respondents own a Holden vehicle. Use this information to construct a 95% confidence interval for the proportion of people in Australia who own a Holden vehicle. State values correct to 3 decimal places.
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15D - Example 1: Video solution
Video coming soon! \[(0.118,0.222)\] 15D - Example 1: Practice
Question 1: ABC Question 2: ABC 15D - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Calculating confidence intervals using CAS:
15D - Example 1: Calculating an approximate 95% confidence interval (CAS)
In Australia, 200 random people were surveyed and it is found the 34 respondents own a Holden vehicle. Use this information to construct a 95% confidence interval for the proportion of people in Australia who own a Holden vehicle. State values correct to 3 decimal places.
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15D - Example 1: Video solution
Video coming soon! \[(0.118,0.222)\] 15D - Example 1: Practice
Question 1: ABC Question 2: ABC 15D - Example 1: Solutions
Question 1: ABC Question 2: ABC |