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AlexFokin [52]
2 years ago
9

g Based on historical data, your manager believes that 32% of the company's orders come from first-time customers. A random samp

le of 146 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is greater than than 0.43
Mathematics
1 answer:
Assoli18 [71]2 years ago
3 0

Answer:

0.0022 = 0.22% probability that the sample proportion is greater than than 0.43

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

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

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean \mu = p and standard deviation s = \sqrt{\frac{p(1-p)}{n}}

Based on historical data, your manager believes that 32% of the company's orders come from first-time customers.

This means that p = 0.32

Mean and standard deviation:

Sample of 146 means that n = 146

\mu = p = 0.32

s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32*0.68}{146}} = 0.0386

What is the probability that the sample proportion is greater than than 0.43?

This is 1 subtracted by the pvalue of Z when X = 0.43. So

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

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{0.43 - 0.32}{0.0386}

Z = 2.85

Z = 2.85 has a pvalue of 0.9978

1 - 0.9978 = 0.0022

0.0022 = 0.22% probability that the sample proportion is greater than than 0.43

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