A Level OCR Hard stats question

I have completed parts a,b and c. But how do i solve d??

You can find the cumulative probability for the left tail
P(X<160)
and add 0.6 to it and take that away from 1. That would give the cumulative probability for the right tail which should be a calculator job to get the required value?

How do I solve part d)

How do I solve part d)

Note the previous edit about sketching the normal / bell curve and marking important values/parameters on.

For d) what is the mean/std dev for the belll curve and roughly where would the min/max lie in relation to those (where on the sketch) so are they symmetric about the mean, roughly how many std dev away from the mean for 1000 data points?

Can you just help me to understand the mark scheme for the answer?

Under the assumptions of the model, what's the probability of a given rock being <112?

From the observed data, what's the probability of a given rock being < 112?

Do these probabilities match up?

Can you just help me to understand the mark scheme for the answer?

Can you please show me how to do this as I think I’m doing something wrong. I am getting p=0.0712 but the markscheme says 0.0708.

Its in the noise, but agree with your value. their numbers correspond to 111.8

Can you please show me how to do this as I think I’m doing something wrong. I am getting p=0.0712 but the markscheme says 0.0708.

Ok so if the value of p(x<112) is greater than p(x<79.6) whcih is the lower limit of 2 standard deviations, then the model isn’t appropriate i.e. for the model to be appropriate p(x<112) should in theory be less than p(x<79.6)???

For me, just thinking about the intervals so for 1000 data points (+/- 3 std dev) with mean=200 and std dev=60 youd expect the min/max values to be about [20,380]. The observed values are very different so a std dev of about 1/2 the models value is ~right.

The +/-2 std devs isnt the best comparison to make here, but its a common one in hypothesis testing so tails of 0.025 each (0.05 in total) and 0.95 in the main body between +/- 2 std devs so [80,320]. Youd expect 25 values less than the value of 80 out of 1000 and 80 is less than 112. The previous paragraph so +/- 3 std devs is the way to go.

Probabilistically, p(x<112)=0.07, which means youd expect 70 values out of the 100 less than this value so the model aint great. Really youd want about 0.0005 in each tail and using that with std dev of 30 gives a min value of 102 which is about right, though the std dev value could be refined if really necessary.

So how can i use this to solve part e?

Ok so if the value of p(x<112) is greater than p(x<79.6) whcih is the lower limit of 2 standard deviations, then the model isn’t appropriate i.e. for the model to be appropriate p(x<112) should in theory be less than p(x<79.6)???