Answer:
B)
Step-by-step explanation:
The middle 70% of all sample means will fall between what two values (lower bound and higher bound)?
You only need to go to the Table of Z and find to 70% the value from Z. I attached this image.
So, the Z-values for middle 70% is equal to (-1.036, 1.036)
We can now make the upper limit and lower limit for the values. That is:
[tex]\alpha = \mu-z*(/frac{sigma}{\sqrt{n})[/tex]
[tex]\alpha_1 = 110 -1.036*5 =104.82[/tex]
[tex]\alpha_2 = 110+1.036*5 =115.18[/tex]
Our interval is (104.8,115.2)
Answer:
B) 104.8 and 115.2
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 normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
[tex]\mu = 110, \sigma = 25, n = 25, s = \frac{25}{\sqrt{25}} = 5[/tex]
So, the middle 70% of all sample means will fall between what two values?
50 - (70/2) = 15th percentile
50 + (70/2) = 85th percentile
15th percentile
X when Z has a pvalue of 0.15. So X when Z = -1.037.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.037 = \frac{X - 110}{5}[/tex]
[tex]X - 110 = -5*1.037[/tex]
[tex]X = 104.8[/tex]
85th percentile
X when Z has a pvalue of 0.85. So X when Z = 1.037.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]1.037 = \frac{X - 110}{5}[/tex]
[tex]X - 110 = 5*1.037[/tex]
[tex]X = 115.2[/tex]
So the correct answer is:
B) 104.8 and 115.2
