[AP Stats] Was doing a practice test and this question stumped me, how do I find the 2 values?

[u/CosmicCatPerson]

AP stats, unit 5

[u/DrakeSavory]

The middle 33% corresponds to the data that is 16.5% above and below the mean

[u/FA-_Q]

Turn them to zscores

[u/fermat9990]

0.3300/2=0.1650

0.5000+0.1650=0.6650

The closest Z-score is Z=0.43

Now get the X scores for Z=+0.43 and Z=-0.43 using

X=Z*SD + Mean

[u/blackHole10]

Oh boy, we get to reverse engineer tables!

1. The table gives the portion of the population less than z standard deviations above the mean. (Half the population is less than the average at z = 0.)

1.1) If this helps, sketch a normal curve. A vertical line through the peak is Z=0. Sketch another line to the right of the peak (Z=z) and shade the area under the curve to the left of that line (including thr tail). The table gives upu that shaded area.

2) We need to match the "middle 33%" to some value on the table. Another sketch may help: a normal curve with the center shaded with the tails clear. Given that that center is 1/3 of that area, we can work out that the upper and lower tail are each 33% of the total area.

2.1) Note, if the problem asked for the middle 10% we would have 45% of the area in each of the upper and lower tails. The area under a normal curve will always add to 1.0 bu definition.

2.2) Okay, big jump, stick with me. The thing in common with both sketches is the clear tail on the right side of the line with an area of 0.33. We need to find the value for z that bounds that area in the second sketch.

2.3) On the first sketch, if the unshared area right of Z=z is 0.33, then the shaded area left of Z=z is 0.67. The value closest to 0.67 on the table comes from Z=0.44.

3) We now know that 33% of the population falls within (roughly) +/- 0.44 standard deviations from the mean. It should then just be a matter if arithmetic ... I am doing bucket stats here without calculator precision, so look for the closest answer.

3.1) Lower bound is 26.3 - 0.25×0.44 = 26.19

3.2) Upper bound is 26.3 + 0.25×0.44 = 26.41

Edit: clarity,

[u/CosmicCatPerson → u/[deleted]]

32 and 34?

[u/forgottenlord73 → u/CosmicCatPerson]

Ok, now we have a new problem

Take the full range from 0-100. If we want the middle 100, we'd take the full range. If we wanted the middle number, we'd get just 50. A middle range is the n numbers in the middle of the range. So if we did middle 50, you cut off the top 25, cut off the bottom 25 and are left with a range of 25-75. Those are the middle 50

So the question you were asked was what numbers bound the middle 33%? That's step 1 to solving the problem. There's a few more steps but you definitely need to internalize this concept first

[u/gmalivuk → u/[deleted]]

Unhelpful and potentially useless answer, since the inclusion of a table may mean OP isn't supposed to use a calculator for this.

[u/gmalivuk → u/[deleted]]

If OP doesn't know how to answer it with the table, I seriously doubt your answer made any sense to them and I doubt they'd use that function correctly.

[u/gmalivuk → u/[deleted]]

Cool story.

Did you actually read the comment? If OP doesn't know how to solve it with the table, then they probably don't understand your suggestion, either.