Exam 6: Continuous Probability Distributions

i. In terms of a formula the standardized value of z is found by z = (X - µ)/ $\sigma$ . ii. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of less than 10. iii. A computed z for X values to the left of the mean is positive.

The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed. What percent of the students scored below 320?

An accelerated life test on a large number of type-D alkaline batteries revealed that the mean life for a particular use before they failed is 19.0 hours. The distribution of the lives approximated a normal distribution. The standard deviation of the distribution was 1.2 hours. About 95.44 percent of the batteries failed between what two values?

One classic use of the normal distribution is inspired by a letter to Dear Abbey in which a wife claimed to have given birth 308 days after a brief visit from her husband, who was serving in the Navy. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. Given this information. (i) the probability of a pregnancy lasting 308 days or longer is 0.0038. (ii. The result suggests that the husband is the father of the child.

For the normal distribution, the mean plus and minus 1.96 standard deviations will include about what percent of the observations?

The average score of 100 students taking a statistics final was 70 with a standard deviation of 7. Assuming a normal distribution, approximately how many scored less than 60?

(i. The proportion of the area under a normal curve to the right of z = -1.21 is 0.8869. (ii) The proportion of the area under a normal curve to the left of z = 0.50 is 0.1914. (iii) The proportion of the area under a normal curve to the left of z = -2.10 is 0.0179.

Tabulation of a strike vote showed that 90% of those voting cast their ballot in favour of strike action. You take a sample of 50 voters. Assume this to be a binomial distribution. What is the probability that fewer than 40 of your sample voters are in favour of strike action?

The mean amount spent by a family of four on food per month is $500 with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month?

A new drug has been developed that is found to relieve nasal congestion in 90 percent of those with the condition. The new drug is administered to 300 patients with the condition. What is the probability that more than 265 will be relieved of nasal congestion?

A statistics student receives a grade of 85 on a statistics midterm. If the corresponding z-score equals +1.5 and the standard deviation equals 7, the average grade on this exam is _______.

Two business major students, in two different sections of economics, were comparing test scores. The following gives the students' scores, class mean, and standard deviation for each section. (i) The student from Section 2 scored better compared to the rest of their section. (ii. The z-score of the student from section 1 is 1.28. (iii) The z-score of the student from section 2 is 2.87.

The driver's seat in most vehicles is set to comfortably fit a person who is at least 159 cm tall. The distribution of heights of adult women is approximately normal with a mean of 161.5 cm and a standard deviation of 6.3 cm. Determine the percentage of women who can be expected to be uncomfortable in the driver's seat of their car without some sort of an adjustment.

A loaf of bread is normally distributed with a mean of 22 ounces and a standard deviation of ½ ounces. (i) The probability that a loaf of bread is 22.25 ounces is 0.0. (ii. The probability that a loaf of bread is > 21 ounces is 0.9772. (iii) The probability that a loaf of bread is < 24 ounces is 1.0.

The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95 percent of the observations lie between what two values?

The standard deviation of any uniform probability distribution is

An analysis of the grades on the first test in History 101 revealed that they approximate a normal curve with a mean of 75 and a standard deviation of 8. The instructor wants to award the grade of A to the upper 10 percent of the test grades. What is the dividing point between an A and a B grade?

What is the area under the normal curve between z = 0.0 and z = 1.79?

i. The normal curve falls off smoothly in either direction from the central value. Since it is asymptotic, the curve gets closer and closer to the X-axis, but never actually touches it. ii. The mean (µ) divides the normal curve into two identical halves. iii. The number of different normal distributions is unlimited.

i. For a normal probability distribution, about 95 percent of the area under normal curve is within plus and minus two standard deviations of the mean and practically all (99.73 percent) of the area under the normal curve is within three standard deviations of the mean. ii. The total area under the normal curve is 1. iii. The mean of a normal probability distribution is 500 and the standard deviation is 10. About 68 percent of the observations lie between 480 and 520.