# confidence interval, distribution and probability

If Z is a standard normal variable, find the probability.

1)

The probability that Z is less than 1.13

2)

The probability that Z is greater than -1.82

3)

P(-0.73 < Z < 2.27)

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 degrees 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 degrees C (denoted by negative numbers) and some give readings above 0 degrees C (denoted by positive numbers). Assume that the mean reading is 0 degrees C and the standard deviation of the readings is 1.00 degrees 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.

4)

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.

Find the indicated probability.

5)

The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week?

Solve the problem.

6)

Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 63.2 inches and 64.0 inches.

Use the confidence level and sample data to find a confidence interval for estimating the population μ.

7)

A group of 51 randomly selected students have a mean score of 25.4 with a standard deviation of 3.1 on a placement test. What is the 90 percent confidence interval for the mean score, μ, of all students taking the test?

8)

A random sample of 144 full-grown lobsters had a mean weight of 18 ounces and a standard deviation of 2.9 ounces. Construct a 98 percent confidence interval for the population mean μ.

Use the margin of error, confidence level, and standard deviation σ to find the minimum sample size required to estimate an unknown population mean μ.

9)

Margin of error: $ 126, confidence level: 99%, σ = $ 534

Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution.

10)

A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 193 milligrams with s = 15.4 milligrams. Construct a 95 percent confidence interval for the true mean cholesterol content of all such eggs.

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#### Solution Preview

For any problem involving the standard normal (Z) distribution, you can use the NORM.S.DIST function in Excel. In the first three problems I show how to do this. You can also get the same results by using a Z-table typically found in the front of your textbook. For any value of a standard normal distribution, say, we can use the NORM.S.DIST function to find the probability that the Z is less than or equal to by typing the following into any cell in an Excel worksheet and pressing <ENTER>

=NORM.S.DIST( ,TRUE)

1)

The probability that Z is less than 1.13

You can either look this up directly in the table in your text, or you can type the following exactly into any call in an Excel worksheet (Don't forget to put the "=" sign in front):

=NORM.S.DIST(1.13,TRUE)

When you hit <Enter>, the value .8708 will appear in the cell. Therefore:

I have included an Excel worksheet where I typed the value 1.13 in cell A3 and in cell B3, I typed the Excel function =NORM.S.DIST(A3,TRUE), you can view the value .8708 in cell B3. I repeated this for the z values in the other problems and got the following results:

2)

The probability that Z is greater than -1.82

Using the same method, I typed the value -1.82 in cell A4 and the value .0344 appeared in cell B4.

From this we get:

3)

P(-0.73 < Z < 2.27)

Again using the same method in cell A5 of the worksheet, I typed -0.73 and in cell A6, I typed 2.27. From this we get:

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 degrees 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 degrees C (denoted by negative numbers) and some give readings above 0 degrees C (denoted by positive numbers). Assume that the mean reading is 0 degrees C and the standard deviation of the readings is ...

#### Solution Summary

The confidence interval, distributions and probability is examined. The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 degrees C at the freezing point of water is determined.