The population proportion is assumed to be 0.16 . If the desired standard error is 0.0127 or less, what is the minimum sample size needed?
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To determine the minimum sample size needed for a population proportion of 0.16 with a desired standard error of 0.0127 or less, you can use the formula for the standard error of proportion: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] Where \( SE \) is the standard error, \( p \) is the population proportion, and \( n \) is the sample size. Setting \( SE \) to 0.0127 and \( p \) to 0.16, you need to solve for \( n \): 1. Plug in the values: \[ 0.0127 = \sqrt{\frac{0.16(1-0.16)}{n}} \] 2. Square both sides: \[ 0.00016129 = \frac{0.16(0.84)}{n} \] 3. Rearranging gives you: \[ n = \frac{0.16(0.84)}{0.00016129} \] 4. Calculate: \[ n = \frac{0.1344}{0.00016129} \approx 833.8 \] Since sample size must be a whole number, round up to 834. Thus, the minimum sample size needed is **834**. So if you're planning to conduct a survey or an experiment based on this population proportion, make sure to gather a sample of at least **834** individuals to achieve your desired level of precision!