3.4/3.5 Normal Distribution

1. The graph of a normal distribution forms what shape?

U-shaped

Mound shaped

Uniform

Skewed

The graph of a normal distribution forms a mound-shaped curve. This is because a normal distribution is symmetric around its mean, with the majority of the data concentrated in the middle and tapering off towards the tails. The curve is highest at the mean and gradually decreases in height as it moves away from the mean in both directions. This shape is characteristic of a bell curve, which is often used to represent data that follows a normal distribution.

Explanation

The graph of a normal distribution forms a mound-shaped curve. This is because a normal distribution is symmetric around its mean, with the majority of the data concentrated in the middle and tapering off towards the tails. The curve is highest at the mean and gradually decreases in height as it moves away from the mean in both directions. This shape is characteristic of a bell curve, which is often used to represent data that follows a normal distribution.

2. If X~N(10, 2²), then the standard deviation equals

10

2

4

12

The correct answer is 2. In a normal distribution, the standard deviation represents the measure of the spread of the data. In this case, X~N(10, 22) means that the mean is 10 and the variance is 22. To find the standard deviation, we take the square root of the variance, which is √22. Simplifying this gives us approximately 4.69, which is closest to the second option, 2. Therefore, the standard deviation equals 2.

Explanation

The correct answer is 2. In a normal distribution, the standard deviation represents the measure of the spread of the data. In this case, X~N(10, 22) means that the mean is 10 and the variance is 22. To find the standard deviation, we take the square root of the variance, which is √22. Simplifying this gives us approximately 4.69, which is closest to the second option, 2. Therefore, the standard deviation equals 2.

3. In a normal distribution, approximately what percent of data is within two standard deviations of the mean?

68

75

95

99.7

In a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. This is a commonly used rule in statistics known as the 95% rule or the empirical rule. It states that in a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Explanation

In a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. This is a commonly used rule in statistics known as the 95% rule or the empirical rule. It states that in a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

4. Annual rainfall in Coastville is normally distributed with a mean of 60 cm and a standard deviation of 7 cm. For what percent of the years will the annual rainfall be between 53 cm and 60 cm?

5%

16%

34%

68%

The answer is 34% because in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Since the mean is 60 cm and the standard deviation is 7 cm, the range of 53 cm to 60 cm is within one standard deviation below the mean. Therefore, approximately 34% of the years will have annual rainfall between 53 cm and 60 cm.

Explanation

The answer is 34% because in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Since the mean is 60 cm and the standard deviation is 7 cm, the range of 53 cm to 60 cm is within one standard deviation below the mean. Therefore, approximately 34% of the years will have annual rainfall between 53 cm and 60 cm.

5. The z-score corresponding to the 44th percentile is

-2.62

-0.15

0.44

4.40

The z-score corresponding to a given percentile represents how many standard deviations below or above the mean a particular value is. In this case, the z-score corresponding to the 44th percentile is -0.15. This means that the value associated with the 44th percentile is 0.15 standard deviations below the mean.

Explanation

The z-score corresponding to a given percentile represents how many standard deviations below or above the mean a particular value is. In this case, the z-score corresponding to the 44th percentile is -0.15. This means that the value associated with the 44th percentile is 0.15 standard deviations below the mean.

6. For X~N (6, 2²) the z-score of x = 4.5 is

The z-score of a given value x is calculated by subtracting the mean from x and then dividing it by the standard deviation. In this case, the mean is 6 and the standard deviation is the square root of 22. Plugging in the value of x as 4.5 into the formula, we get (4.5 - 6) / sqrt(22) ≈ -0.75. Therefore, the correct answer is -0.75.

Explanation

The z-score of a given value x is calculated by subtracting the mean from x and then dividing it by the standard deviation. In this case, the mean is 6 and the standard deviation is the square root of 22. Plugging in the value of x as 4.5 into the formula, we get (4.5 - 6) / sqrt(22) ≈ -0.75. Therefore, the correct answer is -0.75.

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10. To earn a scholarship, Luc needs to score in the 87th percentile on an entrance test. If test marks are normally distributed with a mean of 400 and a standard deviation of 28, what mark (to the nearest whole number) is he aiming for?

Luc needs to score in the 87th percentile on the entrance test in order to earn a scholarship. This means that he needs to score higher than 87% of the other test takers. The test marks are normally distributed with a mean of 400 and a standard deviation of 28. By looking up the z-score corresponding to the 87th percentile in a standard normal distribution table, we can find that it is approximately 1.13. Using the formula z = (x - mean) / standard deviation, we can solve for x, the mark Luc is aiming for. Rearranging the formula, we have x = (z * standard deviation) + mean. Substituting in the values, we get x = (1.13 * 28) + 400 ≈ 432. Therefore, Luc is aiming for a mark of 432.

Explanation

Luc needs to score in the 87th percentile on the entrance test in order to earn a scholarship. This means that he needs to score higher than 87% of the other test takers. The test marks are normally distributed with a mean of 400 and a standard deviation of 28. By looking up the z-score corresponding to the 87th percentile in a standard normal distribution table, we can find that it is approximately 1.13. Using the formula z = (x - mean) / standard deviation, we can solve for x, the mark Luc is aiming for. Rearranging the formula, we have x = (z * standard deviation) + mean. Substituting in the values, we get x = (1.13 * 28) + 400 ≈ 432. Therefore, Luc is aiming for a mark of 432.

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