Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s= \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
Mean: [tex]\mu = 200[/tex]
Standard deviation of the population: [tex]\sigma = 50[/tex]
Size of the sample: [tex]n = 100[/tex]
Standard deviation for the sample mean: [tex]s = \frac{50}{\sqrt{100}} = 5[/tex]
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.
