Q: A hospital reports that the average birth weight is $$6.5$$ lbs with a standard deviation of $$1.4$$ lbs. According to the Empirical Rule, what is the lowest birth weight that would be considered significantly high?

$$6.5 + 2 \times 1.4 = 9.3$$ lbs

Q: A sample of $$500$$ random adult books in a library has an average of $$312$$ pages with a standard deviation of $$26$$ pages. Using the Empirical Rule, what is the central range of page lengths containing $$95\% of $$the books?

$$312 \pm 2 \times 26 = 312 \pm 52$$,\ so\ 260\ to\ 364\ pages$

Q: A student finds that song lengths in a playlist are normally distributed with a mean of $$3.2$$ minutes and a standard deviation of $$0.7$$ minutes. What is the range of song lengths that includes approximately $$99.7\% of $$all songs?

$$3.2 \pm 3 \times 0.7 = 1.1\ to\ 5.3$$ minutes

Q: A restaurant gives a coupon to the top $$2.5\% of $$wait times. If the mean wait time is $$4$$ minutes and the standard deviation is $$2.5$$ minutes, what is the minimum wait time to get a coupon?

$$4 + 2 \times 2.5 = 9$$ minutes

Q: A sample of $$500$$ books has a mean of $$312$$ pages and a standard deviation of $$26$$ pages. What is the range of pages that would be considered not significant according to the Empirical Rule?

$$312 \pm 2 \times 26 = 260\ to\ 364$$ pages

Question 1

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

$$95\%$$

Question 2

A hospital reports that the average birth weight is $$6.5$$ lbs with a standard deviation of $$1.4$$ lbs. According to the Empirical Rule, what is the lowest birth weight that would be considered significantly high?

Accepted Answer

$$6.5 + 2 	imes 1.4 = 9.3$$ lbs

Question 3

A sample of $$500$$ random adult books in a library has an average of $$312$$ pages with a standard deviation of $$26$$ pages. Using the Empirical Rule, what is the central range of page lengths containing $$95\% of $$the books?

Accepted Answer

$$312 \pm 2 	imes 26 = 312 \pm 52$$,\ so\ 260\ to\ 364\ pages$

Question 4

The mean wait time at a restaurant is $$4$$ minutes with a standard deviation of $$2$$ minutes $$30$$ seconds. What percentage of wait times are expected to be between $$1.5$$ minutes and $$6.5$$ minutes?

Accepted Answer

$$68\%$$

Question 5

If a data set is approximately normal, which of the following statements is TRUE according to the Empirical Rule?

Accepted Answer

About $$68\% of $$the data falls within $$1$$ standard deviation of the mean.

Question 6

A student finds that song lengths in a playlist are normally distributed with a mean of $$3.2$$ minutes and a standard deviation of $$0.7$$ minutes. What is the range of song lengths that includes approximately $$99.7\% of $$all songs?

Accepted Answer

$$3.2 \pm 3 	imes 0.7 = 1.1\ to\ 5.3$$ minutes

Question 7

In a normal distribution, what is the probability that a randomly selected value is more than $$2$$ standard deviations above the mean?

Accepted Answer

$$2.5\%$$

Question 8

A restaurant gives a coupon to the top $$2.5\% of $$wait times. If the mean wait time is $$4$$ minutes and the standard deviation is $$2.5$$ minutes, what is the minimum wait time to get a coupon?

Accepted Answer

$$4 + 2 	imes 2.5 = 9$$ minutes

Question 9

Which of the following best describes the Empirical Rule for a normal distribution?

Accepted Answer

It estimates the percentage of data within $$1$$, $$2$$, and $$3$$ standard deviations of the mean.

Question 10

A company finds the mean weight of milk bottles is $$12 oz $$with a standard deviation of $$0.5 oz. $$What weight would be considered significantly low?

Accepted Answer

$$12 - 2 	imes 0.5 = 11 oz$$

Question 11

If a data set is not normally distributed, which of the following is a limitation of using the Empirical Rule?

Accepted Answer

The percentages predicted by the Empirical Rule may not accurately reflect the actual data distribution.

Question 12

A sample of $$250$$ wait times at a restaurant has a mean of $$4$$ minutes and a standard deviation of $$2.5$$ minutes. What percentage of wait times are expected to be greater than $$6.5$$ minutes?

Accepted Answer

$$16\%$$

Question 13

A sample of $$500$$ books has a mean of $$312$$ pages and a standard deviation of $$26$$ pages. What is the range of pages that would be considered not significant according to the Empirical Rule?

Accepted Answer

$$312 \pm 2 	imes 26 = 260\ to\ 364$$ pages

Question 14

If the mean wait time at a restaurant is $$4$$ minutes and the standard deviation is $$2.5$$ minutes, what is the percentage of wait times between $$1.5$$ and $$6.5$$ minutes?

Accepted Answer

$$68\%$$

Question 15

A student wants to estimate the standard deviation of song lengths in a playlist. The mean is $$3.2$$ minutes, and the range of most song lengths is from $$2.3$$ minutes to $$4.7$$ minutes. What is the estimated standard deviation?

Accepted Answer

$$(4.7 - 2.3) / 4 = 0.6$$ minutes

Empirical Rule and Standard Deviation: Interpretation and Applications

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