Notes 8.2: Confidence Intervals with Proportions



Learning Target Predict Proportion Parameter (p) with Statistics (p?) 8.2 Learning Target Confidence Intervals with ProportionsIt is important that you use the proper symbols for the variables that you are given and you are solving for.STATISTICS PARAMETERS MEAN ST. DEV. PROPORTIONSEstimation: Using your statistics to predict parameters (Confidence Intervals)Problem…How wide should the interval be? That depends upon how much confidence you want in the estimate.For instance, say you wished to give a confidence interval for the mean income of a college graduate.You might have:that the mean income of a college grad is between100% confidence$0 and $∞95% confidence$35,000 and $41,00090% confidence$36,000 and $40,00080% confidence$37,500 and $38,5000% confidence$38,000 (a point estimate)The wider the interval, the greater the confidence you have that your answer contains the population parameter. However, the wider the interval, the less precision you will have.There is always a struggle between confidence and precision. The more confidence, the more spread…but less precise!469519011366500To estimate population parameters (? or p), we use sample statistics (x? or p?). In a normal distribution:68% of samples fall between ±1 SD 95% of samples fall between ±2 SD 99.7% of samples fall between ±3 SD You can use these values to estimate the value of a parameter, however there is a formula since most confidence intervals are not estimated at 68%, 95%, or 99.7 %.? I can find values of z* based on confidence levels ?In order to estimate at any percent, we need to find the value for z* (z-star, called the upper critical value for z).The z* is a measure of the number of standard deviations away from the mean you would be if you wanted to be within a certain percent. Use the table to find the value for z* based on the confidence interval you are interested in:Example 1: Find the z* values for the following confidence intervals: 90%95%80%Values for z* make sense when the data is approximately normal. To check to see if our sampling distribution is approximately normal, we check these conditions: Conditions for normal distributions with proportions: n p? > 10 and n (1 – p?) > 10When both of these values are at least 10, then the data is approximately normal.? I can calculate the confidence interval for proportions ? Confidence Interval = Statistic + (critical value) (standard deviation of statistic) = p? + (z*) (standard deviation of statistic)The formula for the standard deviation is calculated based on the sample size. The larger the sample size, the smaller the spread. When you look at the formula, this should make sense (because the standard deviation is divided by n). Standard deviation of the statistic = p? ( 1-p?)n The standard deviation that is calculated with a statistic is also called the standard error.To do a confidence interval, follow the steps: (ICCI)I: Identify. State the statistic that you have and the parameter you are trying to predictC: Check conditions. Was the data from a random sample (no bias?)Check for normal: n p? > 10 and n (1 – p?) > 10. C: Calculate. Find:Standard deviation using the formulaz* from tableConfidence interval using the formulaI: Interpret in context. Write a sentence using the words that describe what the problem was about.Example 2: Car manufacturers are interested in customer loyalty- the proportion of customers who would buy another car by that manufacturer. Suppose Chevrolet would like to estimate the true proportion of customer loyalty with 96% confidence. They survey 200 people and find that 160 would buy another Chevrolet.Identify: What statistic do you have and what are you trying to predict?Conditions: Check for a non-bias data and a normal distribution (use formula)Calculate: Standard Deviation for p? (use formula) Find z* value for the 96% confidence interval: ______________ Confidence Interval to predict p (use formula)Interpret: Using a 96% confidence interval, I predict the true proportion of Chevy customers that would buy another Chevrolet is from _________________ to ______________________ .? I can calculate and explain margin of error for proportions ?The confidence interval equation using the statistic is (p?) + (z*)(standard deviation of statistic).Margin of Error: Just the “+ “ part of the equation This is the amount of “error” in either direction, that’s why the + What formula would you use if you were looking for the margin of error (instead of the entire interval)? Margin of error (the + part) = Example 3: Car manufacturers are interested in customer loyalty- the proportion of customers who would buy another car by that manufacturer. Suppose Chevrolet would like to estimate the true proportion of customers who would buy another Chevrolet. They survey 200 people and find that 160 would buy another Chevrolet. Find the margin of error at a 90% confidence.Find the margin of error at a 95% confidence.Find the margin of error at a 99% confidence.As the confidence increases, what happens to the margin of error? ................
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