Section 8.3 Estimating a Population Mean In Section 8.3, you’ll learn about: When is known: The one-sample interval for a population mean z σ Choosing the sample size When is unknown: The distributions t σ Constructing a confidence interval for μ Using procedures wisely t Inference about a population proportion usually arises when we study variables. We learned how to categorical construct and interpret confidence intervals for an unknown parameter in . To estimate a population mean, we have p to record values of a variable for a sample of individuals. It makes sense to try to estimate the mean quantitative amount of sleep that students at a large high school got last night but not their mean eye color! In this section, we’ll examine confidence intervals for a population mean μ. When Is Known: The One-Sample z Interval for a Population Mean σ Mr. Schiel’s class did the mystery mean Activity () and got a value of from an SRS of size 16, as shown. The Normal sampling distribution of X for the mystery mean Activity. Their task was to estimate the unknown population mean They knew that the population distribution was Normal μ. and that its standard deviation was = 20. Their estimate was based on the sampling distribution of X. shows this σ Normal sampling distribution once again. To calculate a 95% confidence interval for we use our familiar formula: μ, statistic ± (critical value) · (standard deviation of statistic) The critical = 1.96, tells us how many standardized units we need to go out to catch the middle 95% of the z* sampling distribution. Our interval is We call such an interval a . Whenever the conditions for inference (Random, Normal, Independent) are satisfied and the population standard deviation is known, we can use this method to construct a confidence interval for σ μ. One-Sample Interval for a Population Mean z Draw an SRS of size from a population having unknown mean and known standard deviation As long as the n μ σ. Normal and Independent conditions are met, a level confidence interval for is C μ The critical is found from the standard Normal distribution. z* This method isn’t very useful in practice, however. In most real-world settings, if we don’t know the population mean then we don’t know the population standard deviation either. But we can use the one-sample interval for a z μ, σ
Section 8.3 Estimating a Population Mean In Section 8.3, you’ll learn about: When σ is known: The one-sample z interval for a population mean Choosing the sample size When σ is unknown: The t distributions Constructing a confidence interval for μ Using t procedures wisely Inference about a population proportion usually arises when we study categorical variables. We learned how to construct and interpret confidence intervals for an unknown parameter p in . To estimate a population mean, we have to record values of a quantitative variable for a sample of individuals. It makes sense to try to estimate the mean amount of sleep that students at a large high school got last night but not their mean eye color! In this section, we’ll examine confidence intervals for a population mean μ. When σ Is Known: The One-Sample z Interval for a Population Mean Mr. Schiel’s class did the mystery mean Activity () and got a value of from an SRS of size 16, as shown. The Normal sampling distribution of X for the mystery mean Activity. Their task was to estimate the unknown population mean μ. They knew that the population distribution was Normal and that its standard deviation was σ = 20. Their estimate was based on the sampling distribution of X. shows this Normal sampling distribution once again. To calculate a 95% confidence interval for μ, we use our familiar formula: statistic ± (critical value) · (standard deviation of statistic) The critical z* = 1.96, tells us how many standardized units we need to go out to catch the middle 95% of the sampling distribution. Our interval is We call such an interval a . Whenever the conditions for inference (Random, Normal, Independent) are satisfied and the population standard deviation σ is known, we can use this method to construct a confidence interval for μ. One-Sample z Interval for a Population Mean Draw an SRS of size n from a population having unknown mean μ and known standard deviation σ. As long as the Normal and Independent conditions are met, a level C confidence interval for μ is The critical z* is found from the standard Normal distribution. This method isn’t very useful in practice, however. In most real-world settings, if we don’t know the population mean μ, then we don’t know the population standard deviation σ either. But we can use the one-sample z interval for a
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