Chapter 7: Estimates and Sample Sizes

30, find the sample standard deviation s and use it as an estimate for . 3. Find the critical value that corresponds to the given level of confidence. 4. Find the margin of error E. 5. Form the confidence interval. Example 4: Starting salaries of college graduates who have taken a statistic course: n =28, , the population is normally distributed and . Find a 95% confidence interval for estimating the population mean. Round-off rule for confidence intervals used to estimate : a) If the original set of data in known, round the confidence interval limits to one more decimal place than is used for the original set of data. b) When the original set of data is unknown, round the confidence interval limits to the same number of decimal places used for the sample mean. Example 5: A sample of 54 bears has a mean weight of 182.9 Ib. Assuming that is known to be 121.8 Ib, find a 99% confidence interval estimate of the mean of the population of all such bear weights. Sample size for estimating mean : Given a degree of confidence and a margin of error E, the minimum sample size, n, needed to estimate the population mean Example 6: an economist wants to estimate the mean income for the first year of work for college graduates who have taken a statistic course. How many such incomes must be found if we want to be 95% confident that the sample mean is within $500 of the true population mean? Assume that a previous study has revealed that for such incomes, . Round-off rule for sample size n: when necessary, round up to obtain the next whole number. Estimating a population mean: In many real-life situations, the population standard deviation is unknown. If the random variable is normally distributed (or approximately normally distributed), then we will use a t-distribution. Def: If the distribution of a random variable x is approximately normal, then follows a t-distribution. Critical values of t are denoted by . Several properties of the t-distribution are as follows: 1. The t-distribution is bell-shaped and symmetric about the mean. 2. The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as is calculated. Degrees of 3. 4. 5. 6. freedom = n -1 The total area under a t-curve is 1 or 100%. The mean, median, and mode of the t-distribution are equal to zero. The standard deviation of t distribution varies with the sample size, but it is greater than 1 (unlike the standard normal distribution, which has standard deviation of 1). As the sample size n gets larger, the t-distribution gets closer to the standard normal distribution. A value of can be found in table A-3 by locating the appropriate number of degrees of freedom in the left column and proceeding across the corresponding row until reaching the number directly below the applicable value of for area of two tails. Example 7: find the critical value, for a 95% confidence when the sample size is 15. Example 8: find the critical value, for a 90% confidence when the sample size is 22. Guidelines: Constructing a confidence interval for (with is unknown) (The population appears to be normally distributed or n > 30) 1. Find the sample statistics n and . Sample standard deviation: 2. Identify the degrees of freedom, the level of confidence 1- , and the critical value 3. Find the margin of error E. 4. Form the confidence interval . Example 9: You randomly select 16 restaurants and measure the temperature of the coffee sold at each. The sample mean temperature is 1620 F with a sample standard deviation of 100 F. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed. Example 10: You randomly select 20 mortgage institutions and determine the current mortgage interest rate at each. The sample mean rate is 6.93% with a sample standard deviation of 0.42%. Find the 99% confidence interval for the mean mortgage interest rate. Assume the interest rates are approximately normally distributed. Choosing the Appropriate Distribution: Example 11: determine whether the margin of error E should be calculated using a critical value from the normal distribution, a critical value of from a tdistribution, or neither. a) n = 150, =100, s =15, and the population has a skewed distribution. b) n = 8, =100, s = 5, and the population has a normal distribution. c) n=8, =100, s = 15, and the population has a very skewed distribution d) n = 150, =100, e) n = 8, =100, , and the distribution skewed , and the distribution is extremely skewed.