Answer :
We are given the two equations:
[tex]$$
-1.64 = \frac{60 - \mu}{\sigma} \quad \text{and} \quad 1.96 = \frac{95 - \mu}{\sigma}.
$$[/tex]
Step 1. Express [tex]$\mu$[/tex] in terms of [tex]$\sigma$[/tex] from each equation
From the first equation:
[tex]$$
-1.64 = \frac{60 - \mu}{\sigma} \quad \Longrightarrow \quad 60 - \mu = -1.64\, \sigma,
$$[/tex]
which can be rearranged to:
[tex]$$
\mu = 60 + 1.64\, \sigma.
$$[/tex]
From the second equation:
[tex]$$
1.96 = \frac{95 - \mu}{\sigma} \quad \Longrightarrow \quad 95 - \mu = 1.96\, \sigma,
$$[/tex]
which gives:
[tex]$$
\mu = 95 - 1.96\, \sigma.
$$[/tex]
Step 2. Equate the two expressions for [tex]$\mu$[/tex]
Since both expressions equal [tex]$\mu$[/tex], we set them equal to each other:
[tex]$$
60 + 1.64\, \sigma = 95 - 1.96\, \sigma.
$$[/tex]
Step 3. Solve for [tex]$\sigma$[/tex]
Combine like terms by adding [tex]$1.96\, \sigma$[/tex] to both sides and subtracting 60 from both sides:
[tex]$$
1.64\, \sigma + 1.96\, \sigma = 95 - 60.
$$[/tex]
Simplify both sides:
[tex]$$
3.60\, \sigma = 35.
$$[/tex]
Now, solve for [tex]$\sigma$[/tex]:
[tex]$$
\sigma = \frac{35}{3.60} \approx 9.7222.
$$[/tex]
Step 4. Solve for [tex]$\mu$[/tex]
Substitute the value of [tex]$\sigma$[/tex] into one of the expressions for [tex]$\mu$[/tex]. Using [tex]$\mu = 60 + 1.64\, \sigma$[/tex]:
[tex]$$
\mu = 60 + 1.64 \times 9.7222 \approx 60 + 15.9444 \approx 75.9444.
$$[/tex]
Final Answer
[tex]$$
\mu \approx 75.9444 \quad \text{and} \quad \sigma \approx 9.7222.
$$[/tex]
[tex]$$
-1.64 = \frac{60 - \mu}{\sigma} \quad \text{and} \quad 1.96 = \frac{95 - \mu}{\sigma}.
$$[/tex]
Step 1. Express [tex]$\mu$[/tex] in terms of [tex]$\sigma$[/tex] from each equation
From the first equation:
[tex]$$
-1.64 = \frac{60 - \mu}{\sigma} \quad \Longrightarrow \quad 60 - \mu = -1.64\, \sigma,
$$[/tex]
which can be rearranged to:
[tex]$$
\mu = 60 + 1.64\, \sigma.
$$[/tex]
From the second equation:
[tex]$$
1.96 = \frac{95 - \mu}{\sigma} \quad \Longrightarrow \quad 95 - \mu = 1.96\, \sigma,
$$[/tex]
which gives:
[tex]$$
\mu = 95 - 1.96\, \sigma.
$$[/tex]
Step 2. Equate the two expressions for [tex]$\mu$[/tex]
Since both expressions equal [tex]$\mu$[/tex], we set them equal to each other:
[tex]$$
60 + 1.64\, \sigma = 95 - 1.96\, \sigma.
$$[/tex]
Step 3. Solve for [tex]$\sigma$[/tex]
Combine like terms by adding [tex]$1.96\, \sigma$[/tex] to both sides and subtracting 60 from both sides:
[tex]$$
1.64\, \sigma + 1.96\, \sigma = 95 - 60.
$$[/tex]
Simplify both sides:
[tex]$$
3.60\, \sigma = 35.
$$[/tex]
Now, solve for [tex]$\sigma$[/tex]:
[tex]$$
\sigma = \frac{35}{3.60} \approx 9.7222.
$$[/tex]
Step 4. Solve for [tex]$\mu$[/tex]
Substitute the value of [tex]$\sigma$[/tex] into one of the expressions for [tex]$\mu$[/tex]. Using [tex]$\mu = 60 + 1.64\, \sigma$[/tex]:
[tex]$$
\mu = 60 + 1.64 \times 9.7222 \approx 60 + 15.9444 \approx 75.9444.
$$[/tex]
Final Answer
[tex]$$
\mu \approx 75.9444 \quad \text{and} \quad \sigma \approx 9.7222.
$$[/tex]
