Normal Distribution

Grade 11 · statistics · 68 practice problems · read aloud

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Normal Distribution: The Bell Curve

The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, it appears as the famous "bell curve." 🛎️ It's incredibly useful because many natural phenomena (like heights, test scores, measurement errors) follow this pattern, allowing us to calculate probabilities and make predictions.

Step-by-Step Guide

  1. Identify the Mean (μ) and Standard Deviation (σ): These parameters define your specific normal curve.
  2. Standardize your value (Calculate the z-score): Use the formula: z = (x - μ) / σ. This converts your data point (x) to a standard normal value.
  3. Use the z-table: Find the probability (area under the curve) associated with your z-score. This gives you P(Z < z).
  4. Interpret the probability: Depending on the question, you might need to find the area to the left, right, or between two points.

Worked Examples

Example 1: Probability less than a value

Test scores are normally distributed with μ = 75 and σ = 8. What percentage of students scored less than 80?

  1. z-score: z = (80 - 75) / 8 = 0.625
  2. Use z-table: P(Z < 0.625) ≈ 0.7340
  3. Answer: Approximately 73.4% of students scored below 80.

Example 2: Probability between two values

Using the same test scores (μ=75, σ=8), what percentage scored between 70 and 85?

  1. z-score for 70: z₁ = (70-75)/8 = -0.625 → P(Z < -0.625) ≈ 0.2660
  2. z-score for 85: z₂ = (85-75)/8 = 1.25 → P(Z < 1.25) ≈ 0.8944
  3. Subtract probabilities: 0.8944 - 0.2660 = 0.6284
  4. Answer: Approximately 62.8% of students scored between 70 and 85.

Common Mistakes to Avoid

  • Confusing z-scores with probabilities: A z-score is a standardized value, not a probability. You must use the z-table to convert it.
  • Forgetting the distribution is continuous: P(X ≤ a) is the same as P(X < a) in a continuous distribution.
  • Misinterpreting the z-table: The standard table gives the area to the left of the z-score. For area to the right, remember to subtract from 1.
  • Sign errors on z-scores: A value below the mean will have a negative z-score. Be careful when looking it up!

Tips & Tricks

  • Memorize the 68-95-99.7 Rule: About 68% of data falls within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. This is great for quick estimates!
  • Draw a picture! Always sketch the bell curve, shade the area you're looking for, and mark the mean and your z-scores. A picture prevents logic errors.
  • Check for reasonableness: If your calculated probability is greater than 1 (or 100%), you've made a mistake.

How to Practice

  • Start with simple "less than" problems, then move to "greater than" and "between" problems.
  • Use online applets to manipulate normal curves and see how changing μ and σ affects the shape.
  • Find real-world datasets (like sports statistics or weather data) and see if they are approximately normally distributed.
  • Practice working backwards—finding the data value (x) when given a probability.

Practice problems

6 of the 68, worked through step by step — try them before opening the answer.

1 √(25) + log₂(8) = ?

Hint: Consider the principal square root and the exponent that gives the second number when using base 2

Show the answer

Answer: 8

  1. Calculate the square root: √(25) = 5
  2. Calculate the logarithm: log₂(8) means 2 raised to what power equals 8? Since 2³ = 8, log₂(8) = 3
  3. Add the results: 5 + 3 = 8

The answer is 8.

2 Normal: μ=68, σ=4. What % between 60 and 76?

Hint: Consider how many standard deviations each boundary is from the mean, then apply the empirical rule for normal distributions.

Show the answer

Answer: 95

  1. Calculate how many standard deviations 60 is from the mean: (60 - 68)/4 = -8/4 = -2 standard deviations
  2. Calculate how many standard deviations 76 is from the mean: (76 - 68)/4 = 8/4 = 2 standard deviations
  3. According to the 68-95-99.7 rule for normal distributions, approximately 95% of data falls within 2 standard deviations of the mean
  4. Therefore, approximately 95% of values fall between 60 and 76

The answer is 95.

3 Normal: μ=72, σ=9. What % between 54 and 90?

Hint: Think about how many standard deviations each boundary is from the mean, then recall the empirical rule for normal distributions.

Show the answer

Answer: 95

  1. Calculate the z-score for 54: (54 - 72)/9 = -18/9 = -2.
  2. Calculate the z-score for 90: (90 - 72)/9 = 18/9 = 2.
  3. According to the 68-95-99.7 rule, approximately 95% of data falls within 2 standard deviations of the mean.
  4. Therefore, approximately 95% of values lie between 54 and 90.

The answer is 95.

4 The heights of sunflowers in a field follow a normal distribution with a mean of 185 cm and a standard deviation of 12 cm. What percentage of sunflowers have heights between 161 cm and 209 cm?

Hint: Convert the given heights to z-scores using the mean and standard deviation, then use the empirical rule for normal distributions to find the percentage between these z-scores.

Show the answer

Answer: 95

  1. Calculate the z-score for 161 cm: z = (161 - 185) / 12 = -24 / 12 = -2
  2. Calculate the z-score for 209 cm: z = (209 - 185) / 12 = 24 / 12 = 2
  3. According to the empirical rule for normal distributions, approximately 95% of data falls within 2 standard deviations of the mean (between z = -2 and z = 2).
  4. Therefore, approximately 95% of sunflowers have heights between 161 cm and 209 cm.

The answer is 95.

5 A triangular prism has a right triangle base with legs measuring 6 cm and 8 cm, and a height of 15 cm. The prism is positioned so that the triangular bases are vertical. What is the total surface area of this prism?

Hint: For a triangular prism, the total surface area includes the areas of both triangular bases and the three rectangular lateral faces. When given a right triangle base, you can use the Pythagorean theorem to find the hypotenuse, which becomes important for calculating one of the rectangular faces.

Show the answer

Answer: 408 cm²

  1. Understand the shape** We have a triangular prism with a right triangle base. Legs of the base triangle: 6 cm and 8 cm. Height of prism (length between triangular bases): 15 cm. Triangular bases are vertical — this means the lateral faces are rectangles whose heights are 15 cm. --- **
  2. Find the hypotenuse of the triangular base** Right triangle legs: 6 cm and 8 cm. Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm. --- **
  3. Find the area of one triangular base** Area of triangle = (1/2) × base × height (legs are perpendicular) = (1/2) × 6 × 8 = 24 cm². There are 2 triangular bases: total base area = 2 × 24 = 48 cm². --- **
  4. Find the lateral surface area** Lateral surface area = perimeter of base triangle × height of prism. Perimeter of base triangle = 6 + 8 + 10 = 24 cm. Lateral surface area = 24 × 15 = 360 cm². --- **
  5. Total surface area** Total surface area = lateral surface area + area of two triangular bases = 360 + 48 = 408 cm². --- **Final answer:** 408 cm²

Let's go step-by-step. --- **

6 A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A circle is inscribed in this triangle such that it is tangent to all three sides. What is the radius of this inscribed circle?

Hint: Consider the geometric relationship between a triangle's area, its perimeter, and the radius of its inscribed circle. For a different triangle with vertices at (0,0), (3,0), and (0,4), you would apply the same principle.

Show the answer

Answer: 2

  1. Understand the problem We have a right triangle with vertices at (0,0), (5,0), and (0,12). The legs are along the x-axis and y-axis, so: - Leg along x-axis: length 5 - Leg along y-axis: length 12 - Hypotenuse: from (5,0) to (0,12)
  2. Find the hypotenuse length Using the distance formula: Hypotenuse = sqrt( (5-0)^2 + (0-12)^2 ) = sqrt(25 + 144) = sqrt(169) = 13 So the triangle has sides: 5, 12, 13.
  3. Recall formula for inradius of a right triangle For a right triangle with legs a, b and hypotenuse c, the inradius r is: r = (a + b - c) / 2
  4. Apply the formula a = 5, b = 12, c = 13 r = (5 + 12 - 13) / 2 r = (17 - 13) / 2 r = 4 / 2 r = 2
  5. Conclusion The radius of the inscribed circle is 2. This matches the given correct answer.
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