Respuesta :
Answer:
The machine should be set at a mean weight of 51.23 kg.
Step-by-step explanation:
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.
In this problem, we have that:
[tex]\sigma = 0.75[/tex]
At what mean weight should the machine be set so that only 5% of the bags are underweight contain less than 50kg of sand?
We want 50 to be the 5th percentile.
So when X = 50, Z has a pvalue of 0.05. So when X = 50, Z = -1.645. We use this to find the mean weight [tex]\mu[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{50 - \mu}{0.75}[/tex]
[tex]50 - \mu = -1.645*0.75[/tex]
[tex]\mu = 51.23[/tex]
The machine should be set at a mean weight of 51.23 kg.
