Answer:
[tex]mu=12.1299[/tex]
[tex]\sigma=1.2987[/tex]
Step-by-step explanation:
In the normal distribution curve, we will have 5% below 10 months [horizontal axis] and 75% below 13 months [horizontal axis].
We need to find z-score using z-table [normal table] that corresponds to
5% = 0.05
and
75% = 0.75
Zscore formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where mu is mean and sigma is standard deviation [these are the 2 parameters we are seeking]
So, 0.05 corresponds to z = -1.64, and
0.75 corresponds to z = 0.67
Now we put both of these into z-score formula and solve both equations for mu and sigma.
[tex]-1.64=\frac{10-\mu}{\sigma}[/tex]
and
[tex]0.67=\frac{13-\mu}{\sigma}[/tex]
The first equation becomes:
[tex]-1.64=\frac{10-\mu}{\sigma}\\-1.64\sigma+\mu=10\\mu=10+1.64\sigma\\[/tex]
Now, simplifying 2nd equation and putting this in:
[tex]0.67=\frac{13-\mu}{\sigma}\\0.67\sigma=13-(10+1.64\sigma)\\0.67\sigma=13-10-1.64\sigma\\2.31\sigma=3\\\sigma=1.2987[/tex]
Now finding mu:
[tex]\mu=10+1.64(1.2987)\\\mu=12.1299[/tex]
These two MU(mean) and SIGMA(standard deviation) are the 2 parameters.
The parameters of a normal distribution are the mean and the standard deviation.
From the question, we have:
The 5% of babies means that, the p-value is:
[tex]p = 5\%[/tex] and [tex]x = 10[/tex]
The z score at [tex]p = 5\%[/tex] is:
[tex]z = -1.64[/tex]
Using z formula, we have:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
This gives:
[tex]-1.64 = \frac{10 - \mu}{\sigma}[/tex]
Cross multiply
[tex]-1.64\sigma = 10 - \mu[/tex]
Make the mean the subject
[tex]\mu= 10 +1.64\sigma[/tex]
Also:
The 75% of babies means that, the p-value is:
[tex]p = 75\%[/tex] and [tex]x = 13[/tex]
The z score at [tex]p = 75\%[/tex] is:
[tex]z = 0.67[/tex]
Recall that:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
This gives:
[tex]067 = \frac{13 - \mu}{\sigma}[/tex]
Substitute [tex]\mu= 10 +1.64\sigma[/tex]
[tex]0.67 = \frac{13 - 10 - 1.64\sigma}{\sigma}[/tex]
[tex]0.67 = \frac{3 - 1.64\sigma}{\sigma}[/tex]
Cross multiply
[tex]0.67\sigma = 3 - 1.64\sigma[/tex]
Collect like terms
[tex]0.67\sigma +1.64\sigma= 3[/tex]
[tex]2.31\sigma= 3[/tex]
Divide both sides by 2.31
[tex]\sigma= 1.300[/tex]
Substitute [tex]\sigma= 1.300[/tex] in [tex]\mu= 10 +1.64\sigma[/tex]
[tex]\mu = 10 + 1.64 \times 1.300[/tex]
[tex]\mu = 12.132[/tex]
Hence, the parameter is (12.132, 1.300)
Read more about normal distribution at:
https://brainly.com/question/13759327
