Applying Normal Distribution To Real-World Data Quiz

1) A school recognizes students in the top 10%. What is the minimum qualifying score?

661

662

663

665

90th percentile z ≈ 1.28 → 520 + (1.28×110) ≈ 661

Explanation

90th percentile z ≈ 1.28 → 520 + (1.28×110) ≈ 661

2) A clothing brand designs pants to fit the middle 80% of men by height. What range should they target?

64.0 to 76.0 in

65.2 to 74.8 in

66.2 to 73.8 in

67.0 to 73.0 in

Middle 80% → z = ±1.28 → 70 ± (1.28×3) = 66.2–73.8 in

Explanation

Middle 80% → z = ±1.28 → 70 ± (1.28×3) = 66.2–73.8 in

3) Two friends are 74 inches and 65 inches tall. Who is more unusual relative to the distribution?

74 inches, z = 1.33

74 inches, z = 1.00

65 inches, z = −1.67

65 inches, z = −1.00

|−1.67| > |1.33| → 65 in is more unusual

Explanation

|−1.67| > |1.33| → 65 in is more unusual

4) A basketball team recruits from the tallest 2.5% of men. What is the minimum height?

74.9 in

75.0 in

75.9 in

76.0 in

z = 1.96 → 70 + (1.96×3) ≈ 75.9 in

Explanation

z = 1.96 → 70 + (1.96×3) ≈ 75.9 in

5) What height corresponds to the 25th percentile?

67.0 in

68.0 in

68.3 in

69.0 in

25th percentile z = −0.674 → 70 − (0.674×3) ≈ 68 in

Explanation

25th percentile z = −0.674 → 70 − (0.674×3) ≈ 68 in

6) What proportion of adult males are taller than the doorway?

0.0228

0.0475

0.0668

0.1587

z = (76−70)/3 = 2 → upper tail ≈ 0.0228

Explanation

z = (76−70)/3 = 2 → upper tail ≈ 0.0228

7) If the standard deviation increases to 1.2 mm while the mean stays 100 mm, what is the new proportion above 101.5 mm?

0.0918

0.1056

0.1292

0.1587

z = (1.5/1.2) = 1.25 → area above ≈ 0.1056

Explanation

z = (1.5/1.2) = 1.25 → area above ≈ 0.1056

8) The factory wants at most 0.3% out-of-spec on the high side. What max mean shift (by magnitude) can they tolerate if SD stays 0.8 mm and the upper spec is 101.5 mm?

0.5 mm

0.6 mm

0.7 mm

0.8 mm

z ≈ 2.75 for 0.3% tail → (101.5−mean)/0.8 = 2.75 → shift ≈ 0.7 mm

Explanation

z ≈ 2.75 for 0.3% tail → (101.5−mean)/0.8 = 2.75 → shift ≈ 0.7 mm

9) What proportion of rods are within spec (between 98.5 mm and 101.5 mm)?

0.987

0.980

0.940

0.975

Between z = ±1.875 → ≈ 0.94 of all rods

Explanation

Between z = ±1.875 → ≈ 0.94 of all rods

10) What proportion of rods are shorter than 98.5 mm?

0.0013

0.0030

0.0107

0.0228

z = (98.5−100)/0.8 = −1.875 → area ≈ 0.0228

Explanation

z = (98.5−100)/0.8 = −1.875 → area ≈ 0.0228

11) What is the z-score for a day with a high of 90°F?

1.5

1.8

2.0

2.2

(90−78)/6 = 2.0

Explanation

(90−78)/6 = 2.0

12) The middle 50% of scores are between which two values (approx.)?

520 and 630

484 and 556

484 and 594

446 and 594

Q1, Q3 at z = ±0.674 → 520 ± (0.674×110) ≈ 446–594

Explanation

Q1, Q3 at z = ±0.674 → 520 ± (0.674×110) ≈ 446–594

13) What score marks the top 5%?

675

690

701

701.5

95th percentile z ≈ 1.645 → 520 + (1.645×110) ≈ 701

Explanation

95th percentile z ≈ 1.645 → 520 + (1.645×110) ≈ 701

14) Approximately what percent of students score below 680?

84.1%

89.0%

94.5%

95.0%

For z ≈ 1.6, cumulative area ≈ 94.5%

Explanation

For z ≈ 1.6, cumulative area ≈ 94.5%

15) What is the z-score for a student who scored 680?

1.0

1.2

1.4

1.6

(680−520)/110 ≈ 1.45 ≈ 1.6

Explanation

(680−520)/110 ≈ 1.45 ≈ 1.6

16) Two separate weeks had average highs of 79°F and 83°F with the same day-to-day SD of 6°F. Which week was more unusually warm relative to June’s distribution?

79°F week, z = 0.17

83°F week, z = 0.33

79°F week, z = 0.50

82°F week, z = 0.83

(83−78)/6 = 0.83 → higher z means more unusual

Explanation

(83−78)/6 = 0.83 → higher z means more unusual

17) A restaurant plans extra patio staff on the hottest 10% of days. What temperature threshold should they use?

84.7°F

85.7°F

86.7°F

88.7°F

90th percentile z ≈ 1.28 → 78 + (1.28×6) = 85.7°F

Explanation

90th percentile z ≈ 1.28 → 78 + (1.28×6) = 85.7°F

18) On what percentage of days do highs fall between 72°F and 84°F?

34%

68%

81.5%

95%

Between ±1 standard deviation ≈ 68%

Explanation

Between ±1 standard deviation ≈ 68%

19) What temperature corresponds to the 84th percentile of the distribution?

78°F

82°F

84°F

88°F

84th percentile ≈ z = 1 → 78 + (1×6) = 84°F

Explanation

84th percentile ≈ z = 1 → 78 + (1×6) = 84°F

20) Approximately what proportion of June days have highs above 90°F?

0.0228

0.0359

0.0500

0.0918

For z = 2, the upper-tail area ≈ 0.0228

Explanation

For z = 2, the upper-tail area ≈ 0.0228

Applying Normal Distribution to Real-World Data Quiz

1) A school recognizes students in the top 10%. What is the minimum qualifying score?

2)

What first name or nickname would you like us to use?

2) A clothing brand designs pants to fit the middle 80% of men by height. What range should they target?

3) Two friends are 74 inches and 65 inches tall. Who is more unusual relative to the distribution?

4) A basketball team recruits from the tallest 2.5% of men. What is the minimum height?

5) What height corresponds to the 25th percentile?

6) What proportion of adult males are taller than the doorway?

7) If the standard deviation increases to 1.2 mm while the mean stays 100 mm, what is the new proportion above 101.5 mm?

8) The factory wants at most 0.3% out-of-spec on the high side. What max mean shift (by magnitude) can they tolerate if SD stays 0.8 mm and the upper spec is 101.5 mm?

9) What proportion of rods are within spec (between 98.5 mm and 101.5 mm)?

10) What proportion of rods are shorter than 98.5 mm?

11) What is the z-score for a day with a high of 90°F?

12) The middle 50% of scores are between which two values (approx.)?

13) What score marks the top 5%?

14) Approximately what percent of students score below 680?

15) What is the z-score for a student who scored 680?

16) Two separate weeks had average highs of 79°F and 83°F with the same day-to-day SD of 6°F. Which week was more unusually warm relative to June’s distribution?

17) A restaurant plans extra patio staff on the hottest 10% of days. What temperature threshold should they use?

18) On what percentage of days do highs fall between 72°F and 84°F?

19) What temperature corresponds to the 84th percentile of the distribution?

20) Approximately what proportion of June days have highs above 90°F?