Section 5.2: Confidence Intervals and P-Values - Using Normal Distributions

Section 5.2: Confidence Intervals and P-values - Using Normal Distributions

 Table 5.2 - Normal percentiles for common confidence levels Confidence level 80% 90% 95% 98% 99% 2.32 z* 1.282 1.645 1.960 6 2.576

1) Let’s check with the calculator using InvNorm(area to the left, mean, standard deviation) Since z ~ N(0, 1), we do invNorm(area to the left, 0, 1)

 Central Limit Theorem For random samples with a sufficiently large sample size, the distribution of sample statistics for a mean or a proportion is normally distributed and is centered at the value of the population parameter.

2) Let’s check this out with STATKEY

1  Confidence Interval based on a Normal Distribution If the distribution for a statistic follows the shape of a normal distribution with standard error SE, we find a confidence interval for the parameter using

∗ Sample Statistic ± z ·SE

∗ ∗ ∗ where z is chosen so that the area between −z and +z in the standard normal distribution is the desired level of confidence.

3) Example 1: Obesity in America – Data set 3.2 (year 2010) – page 185, chapter 3 In Chapter 3, we see that the mean BMI (Body Mass Index) for a large sample of US adults is 27.655 lb. per sq. inch. We are told that the standard error for this estimate is 0.009. If we use the normal distribution to find a 99% confidence interval for the mean BMI of US adults: a) What is z*?

b) Find and interpret the 99% confidence interval.

c) Will a 90% confidence interval be longer or shorter? More precise or less precise? Think and answer, then, construct to check.

d) Note: A BMI > 25 is classified as overweight. Based on the constructed intervals, is it plausible that America’s overall average BMI would be classified as overweight? Explain.

2 e) If we test the hypotheses mu = 25, versus mu > 25, what will the conclusion be? Reject Ho or do not reject Ho?

4) Example 2: Obesity in America: Exercises vs Non-exercisers Also in Chapter 3, page 186, we see that the difference in mean BMI between non-exercisers (those who said they had not exercised at all in the last 30 days) and exercisers (who said they had exercised at least once in the last 30 days) is , with a standard error for the estimate of SE = 0.016. If we use the normal distribution to find a 90% confidence interval for the difference in mean BMI between the two groups:

a) What is z*?

b) Find and interpret the 90% confidence interval.

c) Will a 95% confidence interval be longer or shorter? More precise or less precise? Think and answer, then, construct to check.

d) Consider the 95% confidence interval, can we be confident that there is a difference in average BMI between non-exercisers and exercisers? Explain.

e) If we test the hypotheses mu(N) = mu(E) versus mu(N) > mu(E) Based on the interval results, what will the conclusion be?

1. It’s possible for the average BMI of the two groups to be the same. 3 2. Non-exercisers have a higher average BMI than exercisers

Explain your choice.

 Hypothesis Test based on a Normal Distribution

When the distribution of the statistic under H0 is normal, we compute a standardized test statistic using

The p-value for the test is the probability a standard normal value is beyond this standardized test statistic, depending on the direction of the alternative hypothesis.

You will be using (a) The calculator and normalcdf (lower, upper, mean, standard error) to find the p-value. (b) or STATKEY

You will then use methods of chapter 4 to decide whether to reject Ho or do not reject Ho

5) Example 3: Is Divorce Morally Acceptable? In a study introduced in Chapter 4, we learn that 67% of women in a random sample view divorce as morally acceptable. Does this provide evidence that more than 60% of women view divorce as morally acceptable? The standard error for the estimate assuming the null hypothesis is true is 0.021.

(a) Is the variable quantitative or categorical? Is this problem about means or proportions? One group or two groups?

(b) What are the null and alternative hypotheses for this test?

(c) What is the sample statistic?

(d) What is the test statistic?

(e) Use the normal distribution to find the p-value.

4 (f)What is the conclusion of the test?

6) Example 4: Do Men and Women Differ in Opinions about Divorce? In the same study described above, we find that 71% of men view divorce as morally acceptable. Use this and the information in the previous example to test whether there is a significant difference between men and women in how they view divorce. The standard error for the difference in proportions under the null hypothesis that the proportions are equal is 0.029.

(a) Is the variable quantitative or categorical? Is this problem about means or proportions? One group or two groups?

(b) What are the null and alternative hypotheses for this test?

(c) What is the sample statistic?

(d) What is the test statistic?

(e) Use the normal distribution to find the p-value.

(f)What is the conclusion of the test?

5 Section 5.2 - Due next class Name ______

7) Confidence Intervals using the Normal Distribution In a recent survey of 1000 US adults conducted in January 2013, 57% said they dine out at least once per week. The standard error for this estimate is 0.016. Use the normal distribution to find a 95% confidence interval for the proportion of US adults who dine out at least once per week. Interpret your answer.

8) Hypothesis Tests using the Normal Distribution A sample of baseball games shows that the mean length of the games is 179.83 minutes. (The data is given in BaseballTimes). The standard error is 3.75. Does this sample provide evidence that the mean length of time for baseball games is more than 170 minutes? Use the normal distribution and show all details of the test. (Hint: write the hypotheses, sketch, label and shade. Find the test statistic, the p-value and write the conclusion)