Question

"Domestic cats kill many more wild birds in the United States than scientists thought," states a recent article. \({ }^{3}\) Researchers used a sample of \(n=140\) households in the US with cats to estimate that \(35 \%\) of household cats in the US hunt outdoors. (a) Find and interpret a \(95 \%\) confidence interval for the proportion of household cats in the US that hunt outdoors. (b) Is it plausible that the proportion of household cats in the US hunting outdoors is \(0.45 ?\) Is it plausible that it is \(0.30 ?\)

Short answer

The 95% confidence interval estimate for the proportion of household cats in the U.S. that hunt outdoors can be calculated using above steps. The plausibility of a given proportion value, like 0.45 or 0.30, can be determined by checking if it falls within this confidence interval.

Step-by-step solution

Calculate Standard Error

First, calculate the standard error for the proportion. The formula is \(\sqrt{ \frac{p(1-p)}{n} }\), where \(p = 0.35\) represents the sample proportion of household cats that hunt outdoors, and \(n = 140\) is the number of households in the sample. The standard error gives the average distance that the observed values fall from the population mean.

Calculate Confidence Interval

Next, calculate the 95% confidence interval. For this, we need the standard normal value for 95% confidence, which is approximately 1.96. The confidence interval is then \(p \pm z \times SE = 0.35 \pm 1.96 \times SE\). This interval estimates the range in which the true population proportion falls with a 95% confidence level.

Interpret Confidence Interval

If the interval includes a value, it means that this value is plausible for the population proportion given the data we have. If it does not include a value, then that value is considered implausible at the 95% confidence level.

Evaluate Plausibility of Proportion Values

Here, we need to check if the values 0.45 and 0.30 are within the estimated confidence interval. If they fall within the calculated interval, then it is plausible; if not, it is implausible.