Calculate a sample proportion and list its possible values

Calculate a sample proportion

A sample proportion compares the number of successes with the total number of sampled units. Use the success count as the numerator and the sample size as the denominator, then use the requested form

1. Identify the two counts Let x be the number of successes and n be the total number of sampled units \(\displaystyle x=18,\quad n=30\)
2. Divide successes by the sample size Form the sample proportion as x divided by n \(\displaystyle \widehat p=\frac{18}{30}=\frac{3}{5}\)
3. Use an equivalent requested form A reduced fraction, decimal and percentage can represent the same proportion when the question permits them \(\displaystyle \frac{3}{5}=0.6=60\%\)
4. Include every possible boundary For a sample of n units, the success count can be any whole number from zero to n \(\displaystyle n=4\Rightarrow\widehat p\in\left\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\right\}\)

Worked example

A quality check measures 45 devices. Twenty-seven devices pass the stated standard. Find the sample proportion of devices that pass

1. \(\displaystyle x=27,\quad n=45\)The success count is 27 and the sample contains 45 devices
2. \(\displaystyle \widehat p=\frac{x}{n}=\frac{27}{45}\)Put the number of successes over the total sample size
3. \(\displaystyle \frac{27}{45}=\frac{3}{5}=0.6=60\%\)Divide and reduce the fraction, then state equivalent decimal and percentage forms

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