So the assumption of normality must be met (see Chapter 1). The confidence is based on area under a normal curve. We want to estimate the population parameter, such as the mean ( μ) or proportion ( p).We attach a level of confidence to this interval to describe how certain we are that this interval actually contains the unknown population parameter.We use a point estimate (e.g., sample mean) to estimate the population mean.The level of confidence is described as (1- α) * 100%.The compliment to the level of confidence is α (alpha), the level of significance.Level of confidence is expressed as a percent. In the long term, 95% of all samples give an interval that contains µ, the true (but unknown) population mean. In this example, twenty-five samples from the same population gave these 95% confidence intervals. Confidence intervals from twenty-five different samples. Our uncertainty is about whether our particular confidence interval is one of those that truly contains the true value of the parameter.įigure 1.The level of confidence corresponds to the expected proportion of intervals that will contain the parameter if many confidence intervals are constructed of the same sample size from the same population.A confidence interval is an interval of values instead of a single point estimate.We use point estimates to construct confidence intervals for unknown parameters. The sample proportion ( p̂) is the point estimate of the population proportion ( p).The sample mean ( x̄) is a point estimate of the population mean ( μ).This is a single value computed from the sample data that is used to estimate the population parameter of interest. We now want to estimate population parameters and assess the reliability of our estimates based on our knowledge of the sampling distributions of these statistics. Inferences about parameters are based on sample statistics. In the preceding chapter we learned that populations are characterized by descriptive measures called parameters. Where x is the number of elements in your population with the characteristic and n is the sample size. The sample proportion ( p̂) is calculated by With proportions, the element either has the characteristic you are interested in or the element does not have the characteristic. It is just as important to understand the distribution of the sample proportion, as the mean. The population proportion ( p) is a parameter that is as commonly estimated as the mean. Sampling Distribution of the Sample Proportion The Central Limit Theorem tells us that regardless of the shape of our population, the sampling distribution of the sample mean will be normal as the sample size increases. A general rule of thumb tells us that n ≥ 30.So if we do not have a normal distribution, or know nothing about our distribution, the CLT tells us that the distribution of the sample means ( x̄) will become normal distributed as n (sample size) increases.The Central Limit Theorem states that the sampling distribution of the sample means will approach a normal distribution as the sample size increases. Is it normal? What if our population is not normally distributed or we don’t know anything about the distribution of our population? It will have a standard deviation (standard error) equal toīecause our inferences about the population mean rely on the sample mean, we focus on the distribution of the sample mean.The distribution of the sample mean will have a mean equal to µ.As we saw in the previous chapter, the sample mean ( x̄) is a random variable with its own distribution. Typically, we use the data from a single sample, but there are many possible samples of the same size that could be drawn from that population. Inferential testing uses the sample mean ( x̄) to estimate the population mean ( μ).
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