Confidence Intervals



Algebra 2 Name:__________________

Confidence Intervals Notes

Sample Sets of Data

The problem: Imagine that you wanted to find the average height of all the students at Novi High School. There are about 2,000 students who attend this school. Imagine trying to measure each and every student to find the average. What would be an easier and faster way to find the average height? Could you measure a sample, or smaller subset of the population? How many students should there be in your sample? How does the composition of your sample population matter? Imagine your first sample was the boys’ basketball team. Then you sampled the members of student council. How would the means of your two samples differ? Which would be closer to the actual, or true, mean of Novi High School students?

Vocabulary:

Population:

Census of a population:

Sample of a population:

Inferential Statistics:

True Mean:

Sample Mean:

Level of Confidence:

STANDARD ERROR OF THE MEAN: [pic], N = # values in the data set

*The standard error is the ____________________________ of the distribution of the sample. This value measures how well a sample mean estimates the true mean of the population.

*We want the Standard Error of the Mean to be __________________.

*The larger the set of data, the closer [pic] is to the true mean of the population.

Examples:

1. Let’s return to the problem above. Instead of measuring all 2,000 students, you choose a sample of 100 students. The mean height of a sample containing 100 students was 64.2 inches and the standard deviation was 2.7 inches. Determine the standard error of the mean.

|Confidence level 99% 95% 90% 80% 50% |

| Zc 2.58 1.96 1.645 1.28 .6745 |

2. From example 1, the first sample of 100 students has a mean height of 64.2 and standard deviation was 2.7 inches, and the standard error of the mean is 0.27. Determine the range of heights such that the probability is 90% that the mean height of the entire population lies within that range (find a 90% confidence interval).

Formulas for confidence limits:

Lower Limit: Upper Limit:

3. One hundred light bulbs are randomly selected and illuminated. The time for each bulb to burn out is recorded. From this sample, the average life is 350 days with a standard deviation of 45 days. Determine the interval of the sample mean that has 99% level of confidence.

4. One hundred light bulbs are randomly selected and their lifetimes are recorded. From this sample, the average life is 350 days with a standard deviation of 45 days. Determine the range of the same mean that had a 95% level of confidence.

As our confidence level decreased, what happened to our interval?

5. A steel manufacturing firm employs 1500 people. During a given year the mean amount contributes to a charity drive for the homeless was $25.75 per employee with a standard error of the mean of $6.25. What is the probability that a random sample of 25 employees yield a mean between $24.15 and $27.35?

Assignment:

1) The mean weight of 36 boys on the wrestling team is 136.4 pounds and the standard deviation is 4.1 pounds. What is the standard error of the mean?

2) A survey of 225 students indicates that they sleep an average of 7.8 hours per day. The standard deviation for the population is 2.6. Construct and interpret a 95% confidence interval with 7.8 as the estimate of the true mean of the entire population of students.

How would an increase in the standard deviation affect your confidence interval? For example, what would the confidence interval be if [pic]?

3. In a random sample of 100 families in Texas, a TV reporter found that children watch an average of 4.6 hours of TV per day. The standard deviation was 1.4 hours.

a) Find the Standard Error of the Mean. [pic]=

b) Find the 50% confidence interval for the number of hours all children in Texas watch TV.

c) Find the 99% confidence interval for the number of hours all children in Texas watch TV.

d) Which one of the confidence intervals is longer (wider)?

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