Which best describes a population parameter?
Population and ParametersPopulationA population is any large collection of objects or individuals, such as Americans, students, or trees about which information is desired. Show
ParameterA parameter is any summary number, like an average or percentage, that describes the entire population. The population mean \(\mu\) (the greek letter "mu") and the population proportion p are two different population parameters. For example:
The problem is that 99.999999999999... % of the time, we don't — or can't — know the real value of a population parameter. The best we can do is estimate the parameter! This is where samples and statistics come in to play. Samples and statisticsSampleA sample is a representative group drawn from the population. StatisticA statistic is any summary number, like an average or percentage, that describes the sample. The sample mean, \(\bar{x}\), and the sample proportion \(\hat{p}\) are two different sample statistics. For example:
Because samples are manageable in size, we can determine the actual value of any statistic. We use the known value of the sample statistic to learn about the unknown value of the population parameter. Example S.1.1What was the prevalence of smoking at Penn State University before the 'no smoking' policy?The main campus at Penn State University has a population of approximately 42,000 students. A research question is "what proportion of these students smoke regularly?" A survey was administered to a sample of 987 Penn State students. Forty-three percent (43%) of the sampled students reported that they smoked regularly. How confident can we be that 43% is close to the actual proportion of all Penn State students who smoke? Answer
Example S.1.2Are the grades of college students inflated?Let's suppose that there exists a population of 7 million college students in the United States today. (The actual number depends on how you define "college student.") And, let's assume that the average GPA of all of these college students is 2.7 (on a 4-point scale). If we take a random sample of 100 college students, how likely is it that the sampled 100 students would have an average GPA as large as 2.9 if the population average was 2.7? Answer
Example S.1.3Is there a linear relationship between birth weight and length of gestation?Consider the relationship between the birth weight of a baby and the length of its gestation: Answer The dashed line summarizes the (unknown) relationship —\(\mu_Y = \beta_0+\beta_1x\)— between birth weight and gestation length of all births in the population. The solid line summarizes the relationship —\(\hat{y} = \beta_0+\beta_1x\)— between birth weight and gestation length in our random sample of 32 births. The goal of linear regression analysis is to use the solid line (the sample) in hopes of learning about the dashed line (the population). Next... Confidence intervals and hypothesis testsThere are two ways to learn about a population parameter. 1) We can use confidence intervals to estimate parameters.
2) We can use hypothesis tests to test and ultimately draw conclusions about the value of a parameter.
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