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sleet_krkn [62]
2 years ago
14

A population has a mean of 200 and a standard deviation of 50. Suppose a sample of size 100 is selected and x is used to estimat

e m.a.What is the probability that the sample mean will be within ±5 of the population mean?b.What is the probability that the sample mean will be within ±10 of the population mean?
Mathematics
1 answer:
AveGali [126]2 years ago
4 0

Answer:

a) 68% probability that the sample mean will be within ±5 of the population mean.

b) 95% probability that the sample mean will be within ±10 of the population mean.

Step-by-step explanation:

To solve this problem, we have to understand the Empirical Rule and the Central Limit Theorem

Empirical Rule

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

Central Limit Theorem

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

In this problem, we have that:

Mean: \mu = 200

Standard deviation of the population: \sigma = 50

Size of the sample: n = 100

Standard deviation for the sample mean: s = \frac{50}{\sqrt{100}} = 5

a.What is the probability that the sample mean will be within ±5 of the population mean?

5 is one standard deviation from the mean.

By the Empirical Rule, 68% probability that the sample mean will be within ±5 of the population mean.

b.What is the probability that the sample mean will be within ±10 of the population mean?

10 is two standard deviations from the mean

By the Empirical Rule, 95% probability that the sample mean will be within ±10 of the population mean.

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