Question

5.1d.a.4: normal distributions 1
1 suppose two sets of test scores have the same mean, but different standard deviations, \\(\sigma_1\\) and \\(\sigma_2\\), with \\(\sigma_2 > \sigma_1\\). which statement best describes the variability of these data sets?

1. data set one has the greater variability.
2. the variability will be the same for each data set.
3. data set two has the greater variability.
4. no conclusion can be made regarding the variability of either set.

2 the job satisfaction rating at a company is approximately normally distributed with a mean of 12. about 95% of the scores are between 8 and 16. what is the standard deviation of this distribution? justify your answer.
3 the heights of women in the united states are normally distributed with a mean of 64 inches and a standard deviation of 2.75 inches. the percent of women whose heights are between 64 and 69.5 inches, to the nearest whole percent, is

1. 6
2. 68
3. 48
4. 95

4 a manufacturer claims that the number of ounces of a beverage dispensed by one of its automatic dispensers is normally distributed with a mean of 8.0 ounces and a standard deviation of 0.04 ounces. to the nearest tenth of a percent, what percent of the cups filled by this company’s dispenser will contain between 7.9 and 8.11 ounces?

1. 99.5
2. 99.1
3. 99.4
4. 97.6

5 the scores on a collegiate mathematics readiness assessment are approximately normally distributed with a mean of 680 and a standard deviation of 120. determine the percentage of scores between 690 and 900, to the nearest percent.
6 the heights of the members of a ski club are normally distributed. the average height is 64.7 inches with a standard deviation of 4.3 inches. determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall.
7 a population is normally distributed with a mean of 23 and a standard deviation of 1.2. the percentage of the population that falls below 21, to the nearest hundredth, is

1. 0.05
2. 8.29
3. 4.78
4. 91.30

8 the distribution of the diameters of ball bearings made under a given manufacturing process is normally distributed with a mean of 4 cm and a standard deviation of 0.2 cm. what proportion of the ball bearings will have a diameter less than 3.7 cm?

1. 0.0668
2. 0.8664
3. 0.4332
4. 0.9500

9 the mean intelligence quotient (iq) score is 100, with a standard deviation of 15, and the scores are normally distributed. given this information, the approximate percentage of the population with an iq greater than 130 is closest to

1. 2%
2. 48%
3. 31%
4. 95%

Explanation:

Problem 1

Step1: Recall variability and standard deviation

Standard deviation ($\sigma$) measures variability. Larger $\sigma$ means more variability.

Step2: Compare $\sigma_1$ and $\sigma_2$

Given $\sigma_2 > \sigma_1$, so data set with $\sigma_2$ (data set two) has greater variability.

Step1: Recall 95% rule for normal distribution

For normal distribution, about 95% of data is within $\mu \pm 2\sigma$.

Step2: Set up equation

Mean $\mu = 12$, range is 8 to 16. So $12 - 2\sigma = 8$ and $12 + 2\sigma = 16$. Solve $12 - 2\sigma = 8$: $2\sigma = 12 - 8 = 4$, so $\sigma = 2$.

Step1: Calculate z - scores

Mean $\mu = 64$, $\sigma = 2.75$. For $x = 69.5$, $z=\frac{69.5 - 64}{2.75}=\frac{5.5}{2.75}=2$.

Step2: Use normal distribution properties

We want $P(64 < X < 69.5)=P(0 < Z < 2)$ (since $\mu = 64$ is the mean, $z = 0$ at $\mu$). From normal table, $P(Z < 2)=0.9772$, $P(Z < 0)=0.5$. So $0.9772 - 0.5 = 0.4772\approx 48\%$.

Answer:

1. Data set two has the greater variability.

Problem 2