Answer :
Answer:
The mean is 15.93 ounces and the standard deviation is 0.29 ounces.
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.
7% of the bottles containing this soft drink there are less than 15.5 ounces
This means that when X = 15.5, Z has a pvalue of 0.07. So when X = 15.5, Z = -1.475.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.475 = \frac{15.5 - \mu}{\sigma}[/tex]
[tex]15.5 - \mu = -1.475\sigma[/tex]
[tex]\mu = 15.5 + 1.475\sigma[/tex]
10% of them there are more than 16.3 ounces.
This means that when X = 16.3, Z has a pvalue of 1-0.1 = 0.9. So when X = 16.3, Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{16.3 - \mu}{\sigma}[/tex]
[tex]16.3 - \mu = 1.28\sigma[/tex]
[tex]\mu = 16.3 – 1.28\sigma[/tex]
From above
[tex]\mu = 15.5 + 1.475\sigma[/tex]
So
[tex]15.5 + 1.475\sigma = 16.3 – 1.28\sigma[/tex]
[tex]2.755\sigma = 0.8[/tex]
[tex]\sigma = \frac{0.8}{2.755}[/tex]
[tex]\sigma = 0.29[/tex]
The mean is
[tex]\mu = 15.5 + 1.475\sigma = 15.5 + 1.475*0.29 = 15.93[/tex]
The mean is 15.93 ounces and the standard deviation is 0.29 ounces.