Question

algebra 2 hw 5.07 (due 1/15) name 1. when lin drives a golf ball, it travels a mean distance of 200 yards with a standard deviation of 20 yards, and the distribution of lengths is approximately normal. approximately what percentage of her shots will travel less than 180 yards? 2. a distribution is approximately normal with a mean of 18 and a standard deviation of 4. match the interval with the approximate percentage of data that falls within that interval 1. less than 10 a. 2.5% 2. between 6 and 30 b. 13.5% 3. between 10 and 14 c. 16% 4. between 10 and 18 d. 34% 5. between 10 and 26 e. 47.5% 6. between 14 and 22 f. 68% 7. between 18 and 22 g. 95% 8. greater than 22 h. 99.7% 3. tyler looks at this histogram of a distribution with a mean of 6.7 grams and a standard deviation of 1 gram. he claims that approximately 68% of the data is between 5.7 and 7.7 grams. what is the error in his thinking?

Explanation:

---

Problem 1

Step1: Calculate z-score

$z = \frac{x-\mu}{\sigma} = \frac{180-200}{20} = -1$

Step2: Use empirical rule

For $z=-1$, the percentage of data less than this value is $16\%$ (since $100\% - 84\% = 16\%$, as 84% of data is above $z=-1$ in a normal distribution).

---

Problem 2 (Mean $\mu=18$, Std Dev $\sigma=4$)

Step1: Match intervals to z-scores

For each interval, calculate $z=\frac{x-\mu}{\sigma}$, then apply the 68-95-99.7 rule:

1. Less than 10: $z=\frac{10-18}{4}=-2$, percentage = $100\% - 97.5\% = 16\%$ → c
2. Between 6 and 30: $z=\frac{6-18}{4}=-3$, $z=\frac{30-18}{4}=3$, percentage = 99.7% → h
3. Between 10 and 14: $z=-2$ to $z=-1$, percentage = $13.5\%$ → b
4. Between 10 and 18: $z=-2$ to $z=0$, percentage = $34\%$ → d
5. Between 10 and 26: $z=-2$ to $z=2$, percentage = $47.5\%$ (half of 95%) → e
6. Between 14 and 22: $z=-1$ to $z=1$, percentage = $68\%$ → f
7. Between 18 and 22: $z=0$ to $z=1$, percentage = $34\%$ → d
8. Greater than 22: $z=1$, percentage = $100\% - 97.5\% = 2.5\%$ → a

---

Problem 3

Step1: Identify distribution shape

The histogram is right-skewed (tail extends to the right).

Step2: Apply rule constraints

The 68-95-99.7 rule (which states 68% of data is within $\mu\pm\sigma$) only applies to symmetric, normal distributions. Tyler incorrectly applied this rule to a skewed distribution.

Answer:

1. 16%
2. 1. c. 16%, 2. h. 99.7%, 3. b. 13.5%, 4. d. 34%, 5. e. 47.5%, 6. f. 68%, 7. d. 34%, 8. a. 2.5%
3. The 68-95-99.7 rule only applies to normal distributions, but this histogram shows a right-skewed distribution, not a normal one.