Question

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(a) Suppose n = 36 and p = 0.23. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.)
np =
nq =

, p̂ be approximated by a normal random variable because .

What are the values of μp̂ and σp̂? (Use 3 decimal places.)
μp̂ =
σp̂ =

(b) Suppose n = 25 and p = 0.15. Can we safely approximate p̂ by a normal distribution? Why or why not?
, p̂ be approximated by a normal random variable because .

(c) Suppose n = 58 and p = 0.21. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.)
np =
nq =

, p̂ be approximated by a normal random variable because .

What are the values of μp̂ and σp̂? (Use 3 decimal places.)
μp̂ =
σp̂ =

Answer 1

Answer on Question #63722 – Math – Statistics and Probability

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a) Suppose $n = 36$ and $p = 0.23$.

Question

Can we approximate $\hat{p}$ by a normal distribution? Why? (Use 2 decimal places.)

Solution

$$\begin{array}{l} n p = 36 * 0.23 = 8.28 > 5 \\ n q = 36 * 0.77 = 27.22 > 5 \\ \end{array}$$

and $\hat{p}$ can be approximated by a normal random variable, because

$$n p > 5 \text{ and } n q > 5.$$

Answer: yes; because $n p > 5$ and $n q > 5$.

Question

What are the values of $\mu \hat{p}$ and $\sigma \hat{p}$? (Use 3 decimal places.)

Solution

$$\begin{array}{l} \mu \hat{p} = n p = 8.280 \\ \sigma \hat{p} = \sqrt{n p q} = 2.525 \\ \end{array}$$

Answer: 8.28; 2.525.

(b) Suppose $n = 25$ and $p = 0.15$.

Question

Can we safely approximate $\hat{p}$ by a normal distribution? Why or why not?

Solution

$$\begin{array}{l} n p = 25 * 0.15 = 3.75 < 5 \\ n q = 25 * 0.85 = 21.25 > 5 \\ \end{array}$$

and $\hat{p}$ cannot be approximated by a normal random variable because $np < 5$.

**Answer:** no; because $np < 5$.

(c) Suppose $n = 58$ and $p = 0.21$.

Question

Can we approximate $\hat{p}$ by a normal distribution? Why? (Use 2 decimal places.)

Solution

$$\begin{array}{l} n p = 58 * 0.21 = 12.18 > 5 \\ n q = 58 * 0.79 = 45.82 > 5 \\ \end{array}$$

and $\hat{p}$ can be approximated by a normal random variable because

$np > 5$ and $nq > 5$.

**Answer:** yes; because $np > 5$ and $nq > 5$.

Question

What are the values of $\mu \hat{p}$ and $\sigma \hat{p}$? (Use 3 decimal places.)

Solution

$$\begin{array}{l} \mu \hat{p} = n p = 12.180 \\ \sigma \hat{p} = \sqrt{n p q} = 3.102 \\ \end{array}$$

**Answer:** 12.180; 3.102

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