Respuesta :
Answer:
27.58% probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.2 inches
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(which is the square root of the variance) [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 question, we have that:
[tex]\mu = 211, \sigma = \sqrt{2.89} = 1.7, n = 86, s = \frac{1.7}{\sqrt{86}} = 0.1833[/tex]
What is the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.2 inches.
Either greater than 211 + 0.2 = 211.2 or smaller than 211 - 0.2 = 210.8. Since the normal distribution is symmetric, these probabilities are equal, so we find one of them and multiply by 2.
Probability of being less than 210.8:
This is the pvalue of Z when X = 210.8. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{210.8 - 211}{0.1833}[/tex]
[tex]Z = -1.09[/tex]
[tex]Z = -1.09[/tex] has a pvalue of 0.1379
Probability of differing from the population mean by greater than 0.2 inches :
2*0.1379 = 0.2758
27.58% probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.2 inches
Answer:
Probability is 86.24%
Step-by-step explanation:
We can solve this by using the Z-score formula.
Z = (X - μ)/σ
Where;
μ is mean = 211 inches
σ is standard deviation
Now, we are given variance as 2.89
Formula for standard deviation using variance is;
SD = √variance
SD = √2.89
SD = 1.7
To find the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.2 inches, we will find the p-value of z and subtract 1 from it.
Since we are told that the sample shafts would differ by 0.2 inches, thus;
X = 211 + 0.2 = 211.2
Since we are working with sample mean of 86, then we have;
Z = (X - μ)/s
s = σ/√86
s = 1.7/√86
s = 0.1833
So,
Z = (211.2 - 211)/0.1833
Z = 1.0911
From z-score calculator, the p-value is gotten to be 0.1376
Thus, probability that mean diameter of the sample shafts would differ from the population mean by greater than 0.2 inches is;
Probability = 1 - 0.1376 = 0.8624 = 86.24%
