Exam 6: Normal Probability Distributions

Solve the problem. -The weights of certain machine components are normally distributed with a mean of 8.05 g and a standard deviation of 0.09 g. Find the two weights that separate the top 3% and the bottom 3%. Theses weights could serve as limits used to identify which components should be rejected.

Assume that X has a normal distribution, and find the indicated probability. -The mean is $\mu = 60.0$ and the standard deviation is $\sigma = 4.0$ . Find the probability that $X$ is less than $53.0$ .

Solve the problem. -A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 76.

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value greater than 3?

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. - $\text { Find } Q _ { 3 } \text {, the third quartile. }$

Solve the problem. -Human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F. Find the temperature that separates the top 7% from the bottom 93%.

Scores on a test have a mean of 73 and $Q _ { 3 }$ is 80. The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.)

If Z is a standard normal variable, find the probability -The probability that Z lies between -1.10 and -0.36

Provide an appropriate response. -Consider the uniform distribution shown below. Find the probability that x is greater than 6. Discuss the relationship between area under a density curve and probability.

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. -If 7% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others.

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value less than 2.1?

A history teacher assigns letter grades on a test according to the following scheme: A: Top 10% B: Scores below the top 10% and above the bottom 60% C: Scores below the top 40% and above the bottom 20% D: Scores below the top 80% and above the bottom 10% F: Bottom 10% Scores on the test are normally distributed with a mean of 69 and a standard deviation of 13.4. Find the numerical limits for each letter grade.

The diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What is the probability that the diameter of a randomly selected pencil will be less than 0.285 inches?