{"id":1083,"date":"2021-06-11T20:30:02","date_gmt":"2021-06-12T00:30:02","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=1083"},"modified":"2025-06-24T18:34:38","modified_gmt":"2025-06-24T22:34:38","slug":"10-2-distribution-of-the-sample-proportion","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/10-2-distribution-of-the-sample-proportion\/","title":{"raw":"10.2 Distribution of the Sample Proportion","rendered":"10.2 Distribution of the Sample Proportion"},"content":{"raw":"Inferences about the population mean [latex]\\mu[\/latex] are based on the distribution of the sample mean [latex]\\bar{X}[\/latex]. Similarly, inferences about the population proportion [latex]p[\/latex] are based on the distribution of the sample proportion [latex]\\hat{p}[\/latex].\r\n\r\nThe population proportion<\/strong> is defined as\r\n\r\n[latex] p = \\frac{\\text{\\# of individuals having a certain attribute}}{\\text{\\# of individuals in the population}} = \\frac{\\text{\\# of successes}}{N}. [\/latex]\r\n\r\nThe population proportion can be regarded as a special type of population mean if we let the variable of interest be an indicator variable as follows:\r\n

[latex] x_i = \\begin{cases}\r\n1 & \\text{if the ith individual has the attribute (a success)}, \\\\\r\n0 & \\text{if the ith individual does not have the attribute (a failure)}.\r\n\\end{cases}\r\n[\/latex]<\/p>\r\nThen, the population proportion can be rewritten as\r\n\r\n[latex]p=\\frac{\\text{\\# of individuals having a certain attribute}}{\\text{\\# of individuals in the population}} = \\frac{\\text{\\# of successes}}{N} = \\frac{\\sum x_i}{N}.[\/latex]\r\n\r\nThe variable of interest [latex]X[\/latex] has only two possible values: 1 if the individual has the attribute and 0 if not. Randomly select one individual and define [latex]p[\/latex] as the probability that this individual has the attribute. As a result, the probability distribution of [latex]X[\/latex] is\r\n

Table 10.1<\/strong>: Probability Distribution of an Indicator Variable<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n
\r\n
[latex]x[\/latex]<\/em><\/div><\/th>\r\n
\r\n
1<\/div><\/td>\r\n
\r\n
0<\/div><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
\r\n
[latex]P(X=x)[\/latex]<\/em><\/div><\/th>\r\n
\r\n
[latex]p[\/latex]<\/div><\/td>\r\n
\r\n
[latex]1 \u2013 p[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nwith a population mean and population standard deviation:\r\n\r\n[latex]\\mu = \\sum x P(X=x) = 1 \\times p + 0 \\times (1-p) = p,[\/latex]\r\n\r\n[latex]\\sigma = \\sqrt{\\sum x^2 P(X=x) - \\mu^2} = \\sqrt{1^2 \\times p + 0^2 \\times (1-p) - \\mu^2} = \\sqrt{p - p^2} = \\sqrt{p(1-p)}. [\/latex]\r\n\r\nThe sample proportion can be viewed as a special type of sample mean (in the same way that the population proportion can be viewed as a special type of population mean). That is, in a simple random sample of size n<\/em>, the proportion of individuals with the specific attribute is the sample proportion:\r\n

[latex]\\begin{align*} \\hat{p} &= \\frac{\\text{\\# of individuals having a certain attribute in the sample}}{\\text{sample size}}\\\\& = \\frac{\\text{\\# of successes in the sample}}{n} = \\frac{\\sum x_i}{n} = \\bar{x} \\end{align*}[\/latex]<\/p>\r\nwith [latex]x_i = 1[\/latex] if the individual has the attribute and [latex]x_i=0[\/latex] if not.\r\n\r\nRecall from Chapter 6, the sampling distribution of the sample mean [latex]\\bar{X}[\/latex]:\r\n