Thanks fun_jirachi!
Also, I had another question… Sorry, I’m probably being really annoying bringing this up …
But I am confused as to how I am supposed to solve the question below…
Thanks!
Hey! I'm going to assume that you're allowed to use CAS for this question. I'm also going to use 'u' to represent the mean (it looks like mu), and 's' to represent the standard deviation.
The first thing to note is that X is normally distributed. Whenever this happens, I always like to write X ~ N(u, s
2). This reminds us that there are two variables which we can solve for, meaning that we probably need at least 2 equations.
We actually have these 2 equations! They are Pr(X<39.9161)=0.5789 [1] and Pr(X>38.2491)=0.4799 [2]. However, these are really hard to work with since we don't have the mean. This is where we have to convert these equations with z-scores, which provides us with a constant mean and standard deviation. We can then apply the inverse normal distribution CAS thingamajig to proceed. Hopefully these are enough hints to get you going, but I've provided the rest of the working below.
Spoiler
With [1], we can write Pr(Z< (39.9161-u)/s) = 0.5789. With [2], we can write Pr(Z > (38.2491-u)/s) = 0.4799. Note that the second equation should be written with a 'less than' sign because CAS (at least ti-nspire) likes using the 'less than' area. Rewritten, we have Pr(Z < (38.2491-u)/s) = 0.5201.
Now we can apply the CAS inverse normal distribution thingo. We keep the mean as 0 and the standard deviation as 1, but insert the 0.5789 and 0.5201 from above. This outputs two long decimal values, which I will just state as 0.20 (from 0.5789) and 0.05 (from 0.5201) (to 2 d.p.).
Recall how we calculated the z-scores from above in terms of u and s (these were (39.9161-u)/s and (38.2491-u)/s ). These are equivalent to the values obtained from InvNormal above (0.20 and 0.05 respectively). Since this now gives us two equations for two variables, we solve them as simultaneous equations. It would be much faster to solve these equations using CAS. Assuming I didn't mix up any values in my working, we should get s=11.21 (2 d.p.) and u=37.68 (also 2 d.p.).