Confidence Intervals for Means Chapter 23 Confidence Intervals for Means

Topics Confidence Intervals for Means Standard error for means Student’s t-distribution

Confidence Interval A confidence interval gives a range of possible values the mean lies in.

Example A sample of 5 test scores are selected: 83, 90, 72, 95, 85 What is the sample mean? Is this likely to be the actual population mean? How can we improve on this estimate?

Confidence levels We consider confidence levels to be 1 - , so for a 90% CI,  = .10. For an 80% CI,  = .20 For a 99% CI,  = .01 1 -  /2

Example A confidence interval for a population mean is given by (26, 42). 1) Determine the margin of error. 2) Determine the sample mean that was used to find the CI.

Practically Speaking In practice, are we likely to know the population standard deviation? What can we use instead?

The standardized version of We know that when we know the population standard deviation, the standardized version of the sampling distribution of the mean has the standard normal distribution.

The Student’s t-distribution If we used the sample standard deviation instead, we get the studentized version of This is not normal, but it is similar.

Degrees of Freedom When working with the t-distribution, the sample size here greatly effects the distribution. We say that each t-distribution curve has n - 1 degrees of freedom.

Graph of the Student’s t-distribution

Properties of t-curves 1) The total area under a t-curve is 1. 2) A t-curve extends infinitely in both directions, approaching, but never touching the horizontal axis. 3) A t-curve is symmetric about 0. 4) As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve.

Using the t-table (A-53) (Table T in book) As with normal curves, the area under a t-curve between two values is equal to the percentage of data values between these two values. The notation denotes the t-value having the area  to the right under the t-curve. Ex) For a t-curve with df = 12, find Ex) For a t-curve with df = 900, find

Confidence intervals when the population standard deviation is unknown. Assumptions 1) Normal population or large sample 2)  unknown Step 1: For a confidence level of 1 - , use Table T to find with df = n - 1. Step 2: The confidence interval is from

Guidelines for using this procedure We follow the same guidelines as when we know the population standard deviation. For samples of less than 15, employ the procedure if the original data is normal. For samples of 15 to 30, employ the procedure if the data is not badly abnormal or if there are no outliers. Apply regardless of the data if the sample size is above 30.

Margin of Error For a fixed confidence level 1 - , the margin of error (or maximum error of the estimate) is half the length of the confidence interval.

Example A medical researcher wants to investigate the amount of time it takes for patients’ headache pain to be relieved after taking a new prescription painkiller. She plans to use statistical methods to estimate the mean of the population of relief times. The standard deviation is 20 minutes. How large a sample should she take to estimate the mean time to within 1 minute with 90% confidence?

Example The amount of money a family in a certain neighborhood spends on electricity is known to be a normal variable. 10 families are chosen at random with the following electric bills: 75, 62, 48, 90, 95, 73, 45, 85, 80, 75 Find a 90% CI. Find a 99% CI.

Example A pizza house is considering offering the following special: “Pizza delivered in less than 30 minutes or the pizza is free.” Based on a random sample of 40 delivery times, the manager obtains the following 99% confidence interval for the true mean delivery time, : 25 minutes to 30 minutes. The manager concludes that at least 99% of the pizzas are delivered in under 30 minutes and that it will be safe to go ahead with the special offer. Is the manager’s interpretation correct? Why or why not? Explain your thinking.

Confidence and Precision When the sample size remains fixed, increasing the confidence level results in widening the confidence interval (or decreasing the precision.) Conversely, decreasing the confidence level results in narrowing the confidence interval (or increasing the precision.)

Example A sample of 50 bags of dog food reveals a mean weight per bag of 49.87 pounds. The advertised weight is 50 pounds, and the standard deviation is known to be 0.5 pounds. Is this sufficient evidence to suggest that the actual mean weight of these bags is less than 50 pounds per bag?

Example ctd. To answer the question, we perform at a hypothesis test. We will perform the hypothesis test at the 1% significance level. Step 1: State the null and alternative hypotheses. Since we are trying to determine if the weight is lower than advertised, we use a left-tailed test.

Example ctd: Step 2: Decide on the significance level, . We chose  = .01. Step 3: Determine the Critical Values. For a one tailed test, you determine t. In this case, .01 -2.403

Example ctd. Step 4: Determine the value of the test statistic. Step 5: If the value of the test statistic falls in the rejection region, reject the null hypothesis. If the value falls in the non-rejection region, we do not reject the null hypothesis.

Example ctd. Since the test statistic falls in the non-rejection region, we do not reject the null hypothesis, which was that the mean weight of the dog food bags was 50 pounds. Step 6: State the conclusion in words. We can say, that at the 1% significance level, there is not sufficient evidence to reject the null hypothesis, that is, there is not sufficient evidence to suggest that that the manufacturer’s claim is false.

Example Consumer reports indicates that the mean braking distance for a Mercury Sable was 159 feet. Suppose the Sables equipped with tires having a new tread were used in 45 tests. The mean braking distance in these tests is 148 feet. The standard deviation is known to be 23.5 feet. Does this information indicate that the population mean braking distance is improved for the new tire tread?

p-value Test Strategy Step 1: State the null hypothesis and alternative hypothesis. Step 2: Decide on a significance level. We will use  = 0.01.

Example ctd. Step 3: Compute the value of the test statistic. Step 4: Use Table T or Fathom to determine the p-value. In this case, the p-value is .0015 p-value -3.14

Example ctd. Step 5: If , reject the null hypothesis. Otherwise, we do not reject the null hypothesis. Since .0015 < .01, we reject the null hypothesis. Step 6: State the conclusion in words. There is sufficient evidence to suggest that the mean braking distance with the new tire tread is less than 159 feet.

Example A random sample of 71 adult coyotes in a region of northern Minnesota showed the average age to be 2.05 years with a sample standard deviation of .82 years. However, it is thought that the overall population mean age of coyotes is 1.75 years. Does the sample data indicate that coyotes tend to live longer in northern Minnesota? Use a significance level of 0.01.

Example ctd. Step 1: State the null and alternative hypotheses. Step 2: Decide on a significance level. We will use  = 0.01

Example ctd. Step 3: Determine the critical values. Step 4: Determine the value of the test statistic:

Example ctd. Step 5: Since the test statistic lies in the rejection region, we reject the null hypothesis. Also, the P-value is .0014. Step 6: We can conclude, at the 1% significance level that the coyotes in northern Minnesota live longer than the population average.