Normal distribution question

[prisma summit]

q24 thanks guys i have a stats mock tmr💔💔

[lament musk]

so for part a, the sample mean is distributed normally

x̄ ~N(u, 40^2/n)

the next part is asking to find a value n for which P(|x̄ - u| < 15) = 0.95

|x̄-u| < 15

if you divide each side by s

you get

|x̄-u/s| < 15/s -> Z < 15/s

So P(|x̄ - u| < 15) = 0.95 becomes..
P(|Z| < 15/s) = 0.95

the question says that s = 40 but because we're working with the sample mean and not the entire population

variance = 40/n

so s.d = 40/sqrt(n)

P(|Z| < 15/s.d) = 0.95

P(-15/s.d < Z < 15/s.d) = 0.95 --> P(Z > 15/s.d) + P(Z < -15/s.d) = 0.05

1-Φ(15/s.d) + Φ(-15/s.d) = 0.05
1 - Φ(15/s.d) + 1 - Φ(15/s.d) = 0.05
2 - 2Φ(15/s.d) = 0.05
Φ(15/s.d) = 1.95/2
now inverse phi each side
15/s.d = Φ^-1(0.975)
s.d = 15/Φ^-1(0,975)

remember that s.d was 40/sqrt(n)

so

40/sqrt(n) = 15/Φ^-1(0.975)

solve for n and ur done

fackin hell that took so long