Respuesta :
Answer:
0.62% that the sample mean is at most 58. Any probability lower than 5% is considered unusually low, which means that there is evidence to suggest that this high school is substandard.
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 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:
Also important to remember that the standard deviation is the square root of the variance. So
[tex]\mu = 60, \sigma = sqrt{64} = 8, n = 100, s = \frac{8}{\sqrt{100}} = 0.8[/tex]
A specific high school class of n=100 students had a mean score of 58. Is there evidence to suggest that this high school is substandard? (Consider calculating the probability that the sample mean is at most 58 when n=100)
This probability is the pvalue of Z when X = 58. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{58 - 60}{0.8}[/tex]
[tex]Z = -2.5[/tex]
[tex]Z = -2.5[/tex] has a pvalue of 0.0062.
So there is a 0.62% that the sample mean is at most 58. Any probability lower than 5% is considered unusually low, which means that there is evidence to suggest that this high school is substandard.
