# 2nd Midterm Sample Exam

**Topics:**Normal distribution, Probability theory, Standard deviation

**Pages:**14 (4998 words)

**Published:**November 11, 2012

1. A variable X has a distribution which is described by the density curve shown below:

What proportion of values of X fall between 1 and 6? (A) 0.550 (B) 0.575 (C) 0.600 (D) 0.625 (E) 0.650

2. Which of the following statements about a normal distribution is true? (A) The value of µ must always be positive. (B) The value of σ must always be positive. (C) The shape of a normal distribution depends on the value of µ. (D) The possible values of a standard normal variable range from −3.49 to 3.49. (E) The area under a normal curve depends on the value of σ.

3. A variable X follows a uniform distribution, as shown below:

The distribution of X has an interquartile range equal to 4 (since the middle 50% of the data are contained between the values 2 and 6). Consider the variables with the distributions shown below (assume that the heights of the curves are such that they are both valid density curves):

The interquartile range of density curve (I) is of density curve (II) is . (A) (I) less than 4, (II) greater than 4 (B) (I) greater than 4, (II) less than 4 (C) (I) equal to 4, (II) equal to 4 (D) (I) less than 4, (II) less than 4 (E) (I) greater than 4, (II) greater than 4

and the interquartile range

4. A variable X has a uniform distribution on the interval from 2 to 6. The P (4.2 < X < 5.7) is equal to: (A) 0.375 (B) 0.475 (C) 0.575 (D) 0.675 (E) 0.775

5. A variable Z has a standard normal distribution. What is the value b such that P (−0.37 ≤ Z ≤ b) = 0.5749? (A) 2.02 (B) 1.48 (C) 0.97 (D) 0.63 (E) 1.72

6. What is the P (Z > −0.75)? (A) 0.2266 (B) 0.7734 (C) 0.0401 (D) 0.9599 (E) 0.4289

7. A variable X has a normal distribution with mean 100. It is known that about 47.5% of the values of X fall between 85 and 100. What is the approximate value of the standard deviation σ? (A) 5 (B) 7.5 (C) 12.5 (D) 15 (E) 30

8. Suppose that the variable Z follows a standard normal distribution. If P (−b < Z < b) = 0.92, then b is approximately: (A) 1.75 (B) 1.41 (C) 0.82 (D) 1.64 (E) 0.96

The next three questions (9 to 11) refer to the following: The sport of women’s gymnastics consists of four events. Suppose it is known that scores for each event follow a normal distribution with the following means and standard deviations: Event Balance Beam Uneven Bars Vault Floor Exercise Mean Std. Dev. 8.3 0.3 8.6 0.5 8.2 0.4 9.0 0.2

9. What proportion of gymnasts receives a score between 8.2 and 8.7 on the uneven bars? (A) 0.2088 (B) 0.3674 (C) 0.6000 (D) 0.6837 (E) 0.3085

10. The top 6% of gymnasts in each event earn a trip to the national championships. What is the minimum vault score required to make it to the nationals? (A) 8.44 (B) 7.58 (C) 8.90 (D) 8.41 (E) 8.82

11. Julie receives a score of 9.0 on the balance beam, 9.2 on the uneven bars, 9.1 on the vault and a 9.3 on ﬂoor exercise. In which event did Julie do the best relative to other gymnasts? (A) Balance Beam (B) Uneven Bars (C) Vault (D) Floor Exercise (E) Julie did equally well on all events.

12. Speeds of vehicles on a highway follow a normal distribution with mean 106.2 km/h and standard deviation 8.7 km/h. What proportion of vehicles on this highway are travelling above the 100 km/h speed limit? (A) 0.7126 (B) 0.2612 (C) 0.7910 (D) 0.2874 (E) 0.7611

13. The time to complete a particular exam is approximately normally distributed with a mean of 90 minutes and a standard deviation of 10 minutes. What percentage of students will take longer that 95 minutes to complete the exam? (A) 50.00% (B) 69.15% (C) 30.85% (D) 19.15% (E) 22.85%

14. The time it takes skiers to ﬁnish a downhill race follows a normal distribution with mean 58.47 seconds and standard deviation 1.62 seconds. What proportion of skiers ﬁnish the race in exactly 60 seconds? (A) 0.0556 (B) 0.1736 (C) 0.0122 (D) 0.0409 (E) 0.0000

15. Percentage grades in a large Calculus class follow a normal distribution with mean 60 and standard deviation 10....

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