Can someone please help e with this question
1 answer:
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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 and standard deviation
, a large sample size can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
Mean:
Standard deviation of the population:
Size of the sample:
Standard deviation for the sample mean:
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.