A sample of 300 FM residents was taken and 135

Ответ:

morganruhl2

23.01.2022

The 99% confidence interval for the proportion of FM residents whose favorite season is summer is between (0.376, 0.524).

The lower bound of this interval is 0.376.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$ , and a confidence level of $1-\alpha$ , we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$ .

For this problem, we have that:

$n = 300, \pi = \frac{135}{300} = 0.45$

99% confidence level

So $\alpha = 0.01$ , z is the value of Z that has a pvalue of $1 - \frac{0.01}{2} = 0.995$ , so Z = 2.575 .

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.45 - 2.575\sqrt{\frac{0.45*0.55}{300}} = 0.376$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.45 + 2.575\sqrt{\frac{0.45*0.55}{300}} = 0.524$

The 99% confidence interval for the proportion of FM residents whose favorite season is summer is between (0.376, 0.524).

The lower bound of this interval is 0.376.

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