Chapter 9 Estimation using a single sample

What is statistics? -is the science which deals with 1.Collection of data 2.Presentation of data 3.Analysis of data 1.Interpretation of data

Statistical Inference… Statistical inference is the process by which we acquire information and draw conclusions about populations from samples.

Inference… Two types of inference: estimation and hypothesis testing

Population vs Samples Population Parameters – Usually unknown and are estimated by sample statistics using techniques we will learn – Population Mean: μ – Population Standard Deviation: σ – Population Proportion: p Sample Statistics – Used to estimate population parameters – Sample Mean: x̄ – Sample Standard Deviation: s – Sample Proportion: p̂

Estimation… The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic. There are two types of estimators: Point Estimator Interval Estimator

Suppose we wanted to estimate the proportion of blue candies in a VERY large bowl. How might we go about estimating this proportion? We could take a sample of candies and compute the proportion of blue candies in our sample. We would have a sample proportion or a statistic – a single value for the estimate.

1. Point Estimate A single number (a statistic) based on sample data that is used to estimate a population characteristic “point” refers to the single value on a number line. Different samples may produce different statistics. Population characteristic

Example: The paper “The Impact of Internet and Television Use on the Reading Habits and Practices of College Students” investigates the reading habits of college students. The following observations represent the number of hours spent on academic reading in 1 week by 20 college students The dotplot suggest this data is approximately symmetrical. If a point estimate of , the average academic reading time per week for all college students, is desired, an obvious choice of a statistic for estimating  is the sample mean x. However, there are other possibilities – a trimmed mean or the sample median. If a point estimate of , the average academic reading time per week for all college students, is desired, an obvious choice of a statistic for estimating  is the sample mean x. However, there are other possibilities – a trimmed mean or the sample median.

College Reading Continued The mean of the middle 16 observations. So which of these point estimates should we use?

Criteria for Choosing an estimate statistic 1. Choose a statistic that is unbiased (accurate) An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter Unbiased, since the distribution is centered at the true value Biased, since the distribution is NOT centered at the true value

Criteria-I: Unbiased A statistic with mean value equal to the value of the population characteristic being estimated is said to be an unbiased statistic. Sampling distribution of a unbiased statistic Sampling distribution of a biased statistic Original distribution

2. Consistent: Choose a statistic with the smallest standard deviation Unbiased, but has a larger standard deviation so it is not as precise. Unbiased, but has a smaller standard deviation so it is more precise.

Criteria-II Given a choice between several unbiased statistics that could be used for estimating a population characteristic, the best statistic to use is the one with the smallest standard deviation. Unbiased sampling distribution with the smallest standard deviation, the Best choice.

Bias – the level of trustworthiness of a statistic Unbiased Statistic – a statistic whose sampling distribution mean is equal to the true value of the parameter being estimated. Variability (of a statistic) – a description of the spread of the statistic’s sampling distribution

Large bias and large variability. Try Me : Label according to bias and variability.

Small bias and small variability.

Small bias and large variability.

Large bias and small variability.