8.1 Estimating µ with large samples Large sample: n > 30 Error of estimate – the magnitude of the difference between the point estimate and the true parameter. Point estimate – an estimate of a population parameter given by a single number.

Error of estimate We can not say exactly how close x is to µ when µ is unknown In this section we will we use probability to give us an idea of the size of the error of estimate when we use x as a point estimate of µ.

Confidence level The reliability of an estimate will be measured by the confidence level. If we want to find a confidence level of c. We need to find a z value such that the area under the standard normal curve falling between –z c and z c is equal to c.

How “good” is the estimate? An estimate is not very valuable unless we have some measure of how good it is. Remember x is a random variable, so each time we have a different n we can get a different x value. So we will use the Central Limit Theorem: if the same size is large then x has an approximately normal distribution with mean = and standard deviation=

Confidence intervals (large sample)

8.2 Estimating µ with small samples Small samples n < 30 To avoid the error involved in replacing with s, when dealing with small samples, we will introduce a new variable called the student’s t variable.

t- Chart If many random samples of size n are drawn, then we get many t values. These values can be made into a frequency table and then a histogram to give us an idea of the shape of the t-distribution. Degree of freedom – each d.f. gives a different t distribution. If the d.f. is larger than 30, the t distribution and standard normal z distribution are almost the same.

Confidence intervals

8.3 Estimating p in the binomial distribution The binomial distribution is completely determined by the number of trials n and the probability p of success in a single trial.

For most experiments the number of trials is chosen in advance, then the distribution is completely determined by p. We will consider the problem of estimating p under the assumption that n has already been selected. We will assume that the normal curve is a good approximation to the binomial distribution. *remember that np > 5 and nq> 5 must be true

Example Suppose that 800 students are selected at random from a student body of 20,000, they are each given shots to prevent a certain type of flu. All 800 students are exposed to the flu and 600 of them do not get the flu. What is the probability p that the shot will be successful for any single student selected at random from the entire population?

Error of estimate | (p-hat) – p| This is for large samples Mean μ = p Standard deviation σ = √((pq)/n)

Because the distribution of p-hat = r/n is approximately normal we will use features of the standard normal distribution to find bounds for (p-hat) – p. Recall that z c is the number such that an area equal to c under the standard normal curve falls between –z c and z c.

Confidence interval for p

8.4 Choosing the sample size. In design stages of research projects it is a good idea to decide in advance on the confidence level you want and the maximum error of estimate you wish to allow for your project. How you choose depends on Requirements of the project. Practical nature of the problem.

Sample size for estimating μ We must know the value of σ, if the value is unknown we will do a preliminary sampling to approximate.

Example

8.5 Estimating μ 1 - μ 2 and p 1 -p 2 Independent and dependent samples Independent - if the way we take a sample form one population is unrelated to the selection of sample data from the other population. Dependent – if the samples are chosen in such away that each measurement if one sample can be naturally paired with a measurement in the other sample.

Confidence interval of μ 1 - μ 2 (large samples) n is greater than or equal to 30. Confidence interval Max error tolerance

Theorem 8.1 Let x 1 have a normal distribution with mean, __ and standard deviation___. Let x 2 have a normal distribution with mean___ and standard deviation ___. If we take independent random samples of size n 1 from x 1 and n 2 from x 2 then the variable ________ has: 1. A normal distribution 2. Mean ________ 3. Standard deviation_______________

Example

Confidence interval for μ 1 - μ 2 (small samples) If independent random samples are drawn from two populations that possess mean μ 1 and μ 2. We assume that the parent population measurements have normal distributions of close to normal. n is less than 30.

Pooled Variance We estimate the common standard deviation for the two populations by using a pooled variance of s 1 2 and s 2 2 values.

Theorem 8.2 Let x 1 and x 2 have normal distributions with means μ 1 and μ 2 and standard deviation σ 1 and σ 2 respectively. Assume that σ 1 and σ 2 are equal. Suppose we take independent random samples of size n 1 and n 2 from x 1 and x 2. Let x-bar 1 and x-bar 2 and s 1 and s 2 be the sample means and standard deviation.

Theorem 8.2 cont’d If n 1 < 30 and/or n 2 < 30, then

Example

Estimating the difference of proportion p 1 – p 2 Given two binomial probability distributions

Sampling size for estimation p-hat =r/n formulas

Examples

Preliminary estimate for p.

No preliminary estimate for p.

Example