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Exam 8: Continuous Probability Distributions

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Question 51

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A bank has determined that the monthly balances of the saving accounts of its customers are normally distributed, with an average balance of $1200 and a standard deviation of $250. What proportions of the customers have monthly balances: a. less than $1000? b. more than $1125? c. between $950 and $1075?

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a. 0.2119....

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Question 52

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Given that Z is a standard normal random variable, what is the value of Z if the area to the left of Z is 0.1949? $\begin{array}{|l|l|}\hline \text { A. } & 0.51 . \\\hline \text { B. } & -0.51 . \\\hline \text { C. } & 0.86 . \\\hline \text { D. } & -0.86 . \\\hline\end{array}$

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Question 53

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The time it takes a student to complete a 3-hour business statistics sample exam paper is uniformly distributed between 150 and 230 minutes. a. What is the probability density function for this uniform distribution? b. Find the probability that a student will take no more than 180 minutes to complete the sample exam paper. c. Find the probability that a student will take no less than 205 minutes to complete the sample exam paper. d. What is the expected amount of time it takes a student to complete the sample exam paper? e. What is the standard deviation for the amount of time it takes a student to complete the sample exam paper?

The mean of the exponential distribution equals the mean of the Poisson distribution only when the former distribution has a mean equal to: $\begin{array}{|l|l|}\hline\text { A. } & 1.0 . \\\hline \text { B. } & 0.50 \\\hline \text { C. } & 0.25 . \\\hline \text { D. } & \text { any value smaller than } 1.0 \\\hline\end{array}$

Which of the following is a characteristic of a normal distribution? $\begin{array}{|l|l|}\hline A&\text {It is a symmetrical distribution. }\\\hline B&\text {The mean is always zero. }\\\hline C&\text {The mean, median and mode are all equal. }\\\hline D&\text { All of these choices are correct.}\\\hline \end{array}$

Let z₁ be a z-score that is unknown but identifiable by position and area. If the area to the right of z₁ is 0.7291, the value of z₁ is -0.61.

The normal approximation to the binomial distribution works best when the number of trials is large, and when the binomial distribution is symmetrical (like the normal).

If Z is a standard normal random variable, find the value of z that has the following probabilities: a. P(Z ≤ z) = 0.3228. b. P(Z ≥ z) = 0.8289.

If X is a normal random variable with a mean of 78 and a standard deviation of 5, find the following probabilities: a. P(X $\le$ 87). b. P(X $\ge$ 91). c. P( $\le$ X $\le$ ).

Let X be a binomial random variable with n = 100 and p = 0.7. Approximate the following probabilities, using the normal distribution. a. P(X = 75). b. P(X $\le$ 70). c. P(X $\ge$ 60).

Using the standard normal curve, the area between z = 0 and z = 3.50 is about 0.50.

If Z is a standard normal random variable, find the value z for which: a. P(0 $\leq$ Z $\leq$ z) = 0.276. b. P(Z $\geq$ z) = 0.341. c. P(Z $\geq$ z) = 0.819. d. P(-z $\leq$ Z $\leq$ z) = 0.785. e. P(Z $\leq$ z) = 0.9279.

If the random variable X is exponentially distributed with $\lambda$ = 2 parameter, then the variance of the distribution is 0.5.

In the normal distribution, the total area under the curve is equal to one.

Which of the following is always true for all probability density functions of continuous random variables? $\begin{array}{|l|l|}\hline A&\text {They are symmetrical. }\\\hline B&\text { They are bell-shaped.}\\\hline C&\text {The area under the curve is 1.0 . }\\\hline D&\text {They have the same height. }\\\hline \end{array}$

If Z is a standard normal random variable, find the following probabilities. a. P(Z $\leq$ 2.33). b. P(Z $\geq$ 1.65). c. P(-0.58 $\leq$ Z $\leq$ 1.58). d. P(Z $\leq$ -2.27).

In a shopping centre, the waiting time for an elevator is found to be uniformly distributed between 1 and 5 minutes. a. What is the probability density function for this uniform distribution? b. What is the probability of waiting no more than 3 minutes? c. What is the probability that the elevator arrives in the first 30 seconds? d. What is the probability of a waiting time between 2 and 3 minutes? e. What is the expected waiting time?

A random variable X is standardised when each value of X has the mean of X subtracted from it, and the difference is divided by the standard deviation of X.

If the random variable X is exponentially distributed with parameter $\lambda$ = 1.75, then P(1.5 $\le$ X $\le$ 3.8), up to 4 decimal places, is: $\begin{array}{|l|l|}\hline\text { A. } & 0.0711 . \\\hline \text { B. } & 0.0473 . \\\hline \text { C. } & 0.1184 . \\\hline \text { D. } & 0.4739 \\\hline\end{array}$

A fair coin is tossed 500 times. Approximate the probability that the number of tails observed is between 240 and 270 (inclusive).