Naomi weighed 50 patients for a medical study using a scale that measures to the nearest whole pound. She then calculated the mean weight as 176 pounds, with a standard deviation of 12 pounds.

If the distribution is normal, what percent of the data lies between 140 pounds and 212 pounds?

99.7%

95%

34%

68%

To solve this problem, we can use the empirical rule (also known as the 68-95-99.7 rule) for normal distributions. This rule states that for a normal distribution:

1. About 68% of the data falls within 1 standard deviation (±1σ) from the mean.
2. About 95% of the data falls within 2 standard deviations (±2σ) from the mean.
3. About 99.7% of the data falls within 3 standard deviations (±3σ) from the mean.

Here, the mean (\( \mu \)) is 176 pounds, and the standard deviation (\( \sigma \)) is 12 pounds.

- One standard deviation above and below the mean: \( 176 \pm 12 = [164, 188] \)
- Two standard deviations above and below the mean: \( 176 \pm 2 \times 12 = [152, 200] \)
- Three standard deviations above and below the mean: \( 176 \pm 3 \times 12 = [140, 212] \)

The range given in the question is between 140 pounds and 212 pounds, which corresponds to 3 standard deviations from the mean.

According to the empirical rule, **99.7%** of the data lies within 3 standard deviations of the mean. Therefore, the correct answer is:

**99.7%**.

Would you like further details or have any questions?

Here are some related questions for practice:
1. What percentage of the data falls between 152 pounds and 200 pounds?
2. How much data lies below 164 pounds?
3. What is the probability that a randomly selected patient weighs more than 200 pounds?
4. If the weight range was from 164 to 188 pounds, what percent would fall in this interval?
5. How does the standard deviation affect the spread of data in a normal distribution?

**Tip:** The empirical rule provides a quick estimate of how much data lies within 1, 2, or 3 standard deviations of the mean for normally distributed data.

Naomi weighed 50 patients for a medical study using a scale that measures to the nearest whole pound. She then calculated the mean weight as 176 pounds, with a standard deviation of 12 pounds. If the distribution is normal, what percent of the data lies between 140 pounds and 212 pounds? 99.7% 95% 34% 68%

Learn how to determine the percentage of data within a specific range for a normal distribution using the empirical rule. This problem involves a study of 50 patients, with a mean weight of 176 pounds and a standard deviation of 12 pounds. Understand how to apply the 68-95-99.7 rule to find the percent of data between 140 and 212 pounds. Suitable for students in Grades 9-12.

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