Question

In a random sample of 11 residents of the state of New York, the mean waste recycled per person per day was 2.9 pounds with a standard deviation of 0.98 pounds. Determine the 80% confidence interval for the mean waste recycled per person per day for the population of New York. Assume the population is approximately normal. Step 1 of 2 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Answer 1

Answer:

80% confidence interval for the mean waste recycled per person per day for the population of New York is [2.495 , 3.305].

Step-by-step explanation:

We are given that a random sample of 11 residents of the state of New York, the mean waste recycled per person per day was 2.9 pounds with a standard deviation of 0.98 pounds.

Firstly, the pivotal quantity for 80% confidence interval for the true mean waste recycled per person per day for the population of New York is given by;

         P.Q. = $\frac{\bar X - \mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, $\bar X$ = sample mean waste recycled per person per day = 2.9 pounds

             s = sample standard deviation = 0.98 pounds

             n = sample of residents = 11

             $\mu$ = true mean waste recycled per person per day

Here for constructing 80% confidence interval we have used t statistics because we don't know about population standard deviation.

So, 80% confidence interval for the true mean, $\mu$ is ;

P(-1.372 < $t_1_0$ < 1.372) = 0.80  {As the critical value of t at 10 degree of

                                                 freedom are -1.372 & 1.372 with P = 10%}

P(-1.372 < $\frac{\bar X - \mu}{\frac{s}{\sqrt{n} } }$ < 1.372) = 0.80

P( $-1.372 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X - \mu}$ < $1.372 \times {\frac{s}{\sqrt{n} } }$ ) = 0.80

P( $-\bar X -1.372 \times {\frac{s}{\sqrt{n} }$ < - $\mu$ < $-\bar X +1.372 \times {\frac{s}{\sqrt{n} }$ ) = 0.80

P( $\bar X -1.372 \times {\frac{s}{\sqrt{n} }$ < $\mu$ < $\bar X +1.372 \times {\frac{s}{\sqrt{n} }$ ) = 0.80

80% confidence interval for $\mu$ = [ $\bar X -1.372 \times {\frac{s}{\sqrt{n} }$ , $\bar X +1.372 \times {\frac{s}{\sqrt{n} }$ ]

                                                 = [ $2.9 -1.372 \times {\frac{0.98}{\sqrt{11} }$ , $2.9 -1.372 \times {\frac{0.98}{\sqrt{11} }$ ]

                                                 = [2.495 , 3.305]

Therefore, 80% confidence interval for the mean waste recycled per person per day for the population of New York is [2.495 , 3.305].