Question

For the binomial sample sizes and​ null-hypothesized values of p in each​ part, determine whether the sample size is large enough to meet the required conditions for using the normal approximation to conduct a valid​ large-sample hypothesis test of the null hypothesis Upper H 0​: p=p0. Complete:
(1) n = 493, p0 = 0.05

(a) the sample size is not large enough to use the normal approximation.

(b) the sample size is large enough to use the normal approximation.

Answer 1

Answer:

b)The sample size is large enough to use the normal approximation.

Step-by-step explanation:

In one case when sample size is very large usually, the Normal Distribution can be used to calculate an approximate probability of an event. The explanation of this is expained by the Central Limit Theorem which states that when we have a sample size is large, the sampling distribution of means converge to a normal distribution (approximately) and on this way:

$\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})$

The Binomial distribution can be approximated using a Normal Distribution in case when sample size is large. We can consider a sample size is large when we have these two conditions:

np > 10 and n(1-p)>10,

On this case we can assume the random variable

$X \sim Binom(n=493,p_o =0.05)$

If we check the conditions:

np=493*0.05=24.65>10

n(1-p)=493*(1-0.05)=468.35>10

So then we can conclude that b)the sample size is large enough to use the normal approximation.