A random sample is gathered to estimate the percentage of American adults who believe that parents should be required to vaccinate their children for diseases like measles, mumps, and rubella. We know that estimates arising from surveys like that are random quantities that vary from sample-to-sample. In Lesson 8 we learned what probability has to say about how close a sample proportion will be to the true population proportion. Show
In an unbiased random survey sample proportion = population proportion + random error. The Normal Approximation tells us that the distribution of these random errors over all possible samples follows the normal curve with a standard deviation of \[\sqrt{\frac{\text{population proportion}(1-\text{population proportion})}{n}} =\sqrt{\frac{p(1−p)}{n}}\] The random error is just how much the sample estimate differs from the true population value. The fact that random errors follow the normal curve also holds for many other summaries like sample averages or differences between two sample proportions or averages - you just need a different formula for the standard deviation in each case (see sections 9.3 and 9.4 below). Notice how the formula for the standard deviation of the sample proportion depends on the true population proportion p. When we do probability calculations we know the value of p so we can just plug that in to get the standard deviation. But when the population value is unknown, we won't know the standard deviation exactly. However, we can get a very good approximation by plugging in the sample proportion. We call this estimate the standard error of the sample proportion Standard Error of Sample Proportion = estimated standard deviation of the sample proportion = \[\sqrt{\frac{\text{sample proportion}(1-\text{sample proportion})}{n}}\] Recap: the estimated percent of Centre Country households that don't meet the EPA guidelines is 63.5% with a standard error of 3.4%. The Normal approximation tells us that
Thus, a 68% confidence interval for the percent of all Centre Country households that don't meet the EPA guidelines is given by 63.5% ± 3.4% A 95% confidence interval for the percent of all Centre Country households that don't meet the EPA guidelines is given by 63.5% ± 6.8% Note! When you see a margin of error in a news report, it almost always referring to a 95% confidence interval. But other levels of confidence are possible Confidence Intervals for a proportion:For large random samples a confidence interval for a population proportion is given by \[\text{sample proportion} \pm z* \sqrt{\frac{\text{sample proportion}(1-\text{sample proportion})}{n}}\] where z* is a multiplier number that comes form the normal curve and determines the level of confidence (see Table 9.1 for some common multiplier numbers). Table 9.1. Commonly Used Multipliers
Interpreting Confidence IntervalsTo interpret a confidence interval remember that the sample information is random - but there is a pattern to its behavior if we look at all possible samples. Each possible sample gives us a different sample proportion and a different interval. But, even though the results vary from sample-to-sample, we are "confident" because the margin-of-error would be satisfied for 95% of all samples (with z*=2). The margin-of-error being satisfied means that the interval includes the true population value. Properties of Confidence Intervals
What are the conditions for a confidence interval for proportions?There are three conditions we need to satisfy before we make a one-sample z-interval to estimate a population proportion. We need to satisfy the random, normal, and independence conditions for these confidence intervals to be valid.
What are the conditions that must be checked before constructing a confidence interval?Here are the six assumptions you should check when constructing a confidence interval:. Assumption #1: Random Sampling. ... . Assumption #2: Independence. ... . Assumption #3: Large Sample. ... . Assumption #4: The 10% Condition. ... . Assumption #5: The Success / Failure Condition. ... . Assumption #6: Homogeneity of Variances.. What are the 3 assumptions for confidence intervals for the mean?The two categorical samples should be collected randomly or be representative of the population. Data values within each sample should be independent of each other. Data values between the samples should be independent of each other.
What are the conditions for constructing at interval?T interval is good for situations where the sample size is small and population standard deviation is unknown. When the sample size comes to be very small (n≤30), the Z-interval for calculating confidence interval becomes less reliable estimate.
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