How Confident Are You?

If the 90% confidence interval of the mean of a population is given by 45 +/- 3.24, which of the following is correct? A) There is a 90% probability that the true mean is in the interval B) There is a 90% probability that the sample mean is in the interval C) If 1,000 samples of the same size are taken, approximately 900 of the intervals obtained from them will contain the true mean D) There is a 90% probability that a data value, chosen at random, will fall in this interval E) None of these is correct

Statistical Inference provides methods for drawing conclusions about a population based on a sample taken from that population. Two types of inference we will discuss are: 1) confidence intervals 2) hypothesis tests

A confidence interval is a range of numbers used to estimate an unknown population parameter. All confidence intervals are constructed as: estimate + margin of error A confidence level gives the probability that the unknown parameter will be in the given interval if repeated samples are taken.

350,000 high school seniors in California took the SAT last year 350,000 high school seniors in California took the SAT last year. In order to estimate the average SAT math score of all California high school seniors, a SRS of 500 of these students is selected. The average SAT math score of this sample is 461. What could we conclude about the average SAT math score of all California high school seniors?

Central Limit Theorem We can draw an SRS from any population. The population will have mean = μ and standard deviation = σ. When the sample size is large enough, the distribution of the sample averages will be a normal distribution with mean = μ and standard deviation = (σ/√n)

Suppose we know that the standard deviation of SAT math scores for California high school seniors is 100 points. The sampling distribution of SAT math scores for this population would have the following characteristics: Shape: Approximately Normal Mean: 461 Standard Deviation: 100 / √500 or about 4.47

Since the sampling distribution is approximately normal, we can apply the Empirical Rule. 95% of the averages (from sample size 500) will fall in the range 461 + 8.94. The estimate of our interval would be 461, and the margin of error is 8.94. We are 95% confident that the average SAT math score for all California high school seniors is within 8.94 points of 461.

It is appropriate to construct a confidence interval to estimate an unknown parameter µ if the following two conditions are met: The sample is an SRS from the population of interest. The sampling distribution of the sample means is approximately normal.

If the 90% confidence interval of the mean of a population is given by 45 +/- 3.24, which of the following is correct? A) There is a 90% probability that the true mean is in the interval B) There is a 90% probability that the sample mean is in the interval C) If 1,000 samples of the same size are taken, approximately 900 of the intervals obtained from them will contain the true mean D) There is a 90% probability that a data value, chosen at random, will fall in this interval E) None of these is correct