Question

If np>5 and nq>5, estimate P (at least 10) with n=13 and p=0.5 by using the normal distribution as approximation for the binomial distribution, if np, or nq,5, then state that the normal approximation is not suitable.

Answer 1

Using the normal distribution, the probability is given as follows:

$P(X \geq 10) = 0.0485$.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with $\mu = np, \sigma = \sqrt{np(1-p)}$.

The parameters for the binomial distribution are:

p = 0.5, n = 13.

Hence the mean and the standard deviation are:

• $\mu = np = 13 \times 0.5 = 6.5$.

• $\sigma = \sqrt{np(1-p)} = \sqrt{13 \times 0.5 \times 0.5} = 1.8028$

Using continuity correction, the desired probability is P(X > 9.5), which is one subtracted by the p-value of Z when X = 9.5, hence:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{9.5 - 6.5}{1.8028}$

Z = 1.66

Z = 1.66 has a p-value of 0.9515.

1 - 0.9515 = 0.0485, then:

$P(X \geq 10) = 0.0485$.

More can be learned about the normal distribution at brainly.com/question/4079902

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