Notes 8.1: Confidence Intervals with Proportions
Learning Target Predict proportion parameter (p) with statistics (p?) ? I can identify the proportion parameter that I am interested in based on sample data ?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).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%98%96%? I can calculate the confidence interval for proportions ? Confidence Interval = Statistic + (critical value)(standard deviation of statistic) = p? + (z* at given confidence level) (standard deviation of statistic)The formula for the standard deviation is similar to the one we already learned, however we do not know p, so we need to use p? as an estimate for p.Standard deviation of the statistic = p??(1-p??)n Note: The formula is the same as we used in the last unit, except we don’t have p, so we use the p? instead. (When you have the parameter, use it….when you don’t have the parameter, use the statistic.) 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 what you have and what you are trying to predictC: Check conditions. Check for normal (since we don’t have p, use p?): n p? > 10 and n (1 – p?) > 10. C: Calculate. Find:Standard deviation (sketch curve)z*Confidence 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 do you have and what are you looking for?))4124325952500Conditions: (Check for a normal distribution) Calculate: Standard Deviation for p? (put on curve) Find z* value for the 96% confidence interval: ______________ Use the values and plug into the formula to calculate the confidence interval: ______________ Confidence Interval = p? + (z*)(standard deviation of statistic)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: The + part of the equation (z*)(standard deviation of statistic)This is the amount of “error” in either direction (that’s why the + )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?? I can describe relationship between confidence level, margin of error, standard deviation, and n ?Confidence Interval: a spread or range of values. This is used to try to predict the true parameter. Hopefully this interval “captures” the true parameter.Statistic + (critical value)(standard deviation of statistic)p? + z* p? (1-p?)nConfidence Level: The % of the time you hope to “capture” the true parameter. This is determined by z*If the confidence increases, z* _______________, then the margin of error __________________ .Standard Deviation: p(1-p)n If the standard deviation increases, the margin of error ________________ .Sample Size (n): As n increases, the standard deviation _________________, the margin of error ________________ . ................
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