Question

a) in the first set, all three curves have the same mean, 10, but different standard deviations. approximate the standard deviation of each of the three curves.
b) in this set, each curve has the same standard deviation. determine the standard deviation and then each mean.
c) based on these two examples, make a conjecture regarding the effect changing the mean has on a normal distribution. make a conjecture regarding the effect changing the standard deviation has on a normal distribution.

1. the sat verbal test in recent years follow approximately a normal distribution with a mean of 505 ($\mu = 505$) and a standard deviation of 110 ($\sigma = 110$). joey took this test and made a score of 600. the scores on the act english test are approximately a normal distribution with a mean of 17 ($\mu = 17$) and a standard deviation of 2.5 ($\sigma = 2.5$). sarah took this test and made a score of 18. which student made the better score? how do you know?

Explanation:

Step1: Recall properties of normal - distribution

For a normal distribution, about 68% of the data lies within 1 - standard - deviation of the mean, i.e., $\mu\pm\sigma$.

Step2: Approximate standard - deviation for part (a)

The narrowest curve: Most of the data is within 9 to 11. So, $\sigma\approx1$. The middle - width curve: Most of the data is within 8 to 12. So, $\sigma\approx2$. The widest curve: Most of the data is within 6 to 14. So, $\sigma\approx4$.

Step3: Determine standard - deviation and mean for part (b)

Since the curves are symmetric and have the same spread, and they overlap at points 6 and 10. The distance between the intersection points is 4. Since this distance is $2\sigma$ (because of the symmetry of the normal distribution), $\sigma = 2$. The means are the peaks of the curves. The left - hand curve has a mean of 6 and the right - hand curve has a mean of 10.

Step4: Make conjectures for part (c)

Changing the mean shifts the normal distribution along the x - axis. Changing the standard deviation changes the spread of the normal distribution. A smaller standard deviation makes the curve narrower and taller, while a larger standard deviation makes the curve wider and flatter.

Step5: Compare scores for part (4)

Calculate the z - scores. The z - score formula is $z=\frac{x-\mu}{\sigma}$.
For Joey (SAT): $\mu = 505$, $\sigma=110$, $x = 600$. Then $z_{Joey}=\frac{600 - 505}{110}=\frac{95}{110}\approx0.86$.
For Sarah (ACT): $\mu = 17$, $\sigma = 2.5$, $x = 18$. Then $z_{Sarah}=\frac{18 - 17}{2.5}=\frac{1}{2.5}=0.4$.
Since $z_{Joey}>z_{Sarah}$, Joey made a better score relative to the distribution of scores on his test.

Answer:

a) Narrowest curve: $\sigma\approx1$, Middle - width curve: $\sigma\approx2$, Widest curve: $\sigma\approx4$.
b) $\sigma = 2$, Left - hand curve mean: 6, Right - hand curve mean: 10.
c) Changing the mean shifts the normal distribution along the x - axis. Changing the standard deviation changes the spread of the normal distribution. A smaller standard deviation makes the curve narrower and taller, while a larger standard deviation makes the curve wider and flatter.
d) Joey made a better score since $z_{Joey}\approx0.86$ and $z_{Sarah}=0.4$ and $z_{Joey}>z_{Sarah}$.