Respuesta :
Answer:
0.0802 = 8.02% probability that the sample mean would differ from the population mean by more than 2.2 millimeters
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 normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, 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.
Their current steel bolts have a mean diameter of 148 millimeters, and a standard deviation of 7 millimeters.
This means that [tex]\mu = 148, \sigma = 7[/tex]
Sample of 31:
This means that [tex]n = 31, s = \frac{7}{\sqrt{31}}[/tex]
What is the probability that the sample mean would differ from the population mean by more than 2.2 millimeters?
Less than 148 - 2.2 = 145.8 or more than 148+2.2 = 150.2. Due to the simmetry of the normal distribution, these probabilities are the same, which means that we find one and multiply by 2.
Probability of being less than 145.8:
The pvalue of Z when X = 145.8. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{145.8 - 148}{\frac{7}{\sqrt{31}}}[/tex]
[tex]Z = -1.75[/tex]
[tex]Z = -1.75[/tex] has a pvalue of 0.0401
2*0.0401 = 0.0802
0.0802 = 8.02% probability that the sample mean would differ from the population mean by more than 2.2 millimeters
