Provide Information Regarding Statistics & Econometrics : Statistical Inferences & Estimation Lecture 26

Statistical Inference

Lecture 26

Generally speaking, inference is the process of drawing logical conclusions from known or presumed to be true premises. The inference made is also known as an idiomatic.

Definition: a conclusion derived from logic and supporting data.

Inference may be divided into two:

Inductive inference
Deductive inference

1. Inductive inference: The induction approach is “bottom-up” in nature. Conclusion is drawn about general on the basis of specific.

2. Deductive inference: Deduction employs a "top-down" methodology. On the basis of general. a specific conclusion is reached.

Statistical Inference

The process of deriving conclusions about population based on data given a sample of the population. There are two main types of statistical inference.

1. Estimation

2. Hypothesis Testing

Estimation

The procedure by which we used sample data selected from the population to estimate the true but unknown value of a parameter. For instance, calculating the sample mean and sample variance of a sample taken from a population. The sample mean and sample variance are the estimates of population mean and variance.

There are two types of estimation.

i. Point Estimation

The estimate is referred to as a point estimate if the estimated value of the unknown may be stated by a single value, and the process is called point estimation.

Example: A middle-class family's monthly expenses average Rs 45000.

ii. Interval Estimation

Interval estimation is the process wherein the estimated value of an unknown parameter is stated by two limits, say a and b, where a < b. This type of estimate is known as an interval estimate.

As an example, a middle-class family's monthly costs typically range from Rs 45,000 to Rs 60,000 per month.

Estimate

The numerical value of the unknown parameter obtained from sample data is called an estimate. The symbol θ is customarily used to denote an unknown parameter, while an estimate of θ is commonly denoted by θ^.

Estimator

The rule or procedure by which an estimate is obtained.

Properties for Satisfactory Point Estimator

A point estimator is considered a good estimator if it possesses the following properties:

1. Unbiasedness

An estimate is said to be unbiased if the statistic used as an estimate has its expected value equal to the true value of the parameter being estimated.

Let θ^ be the estimate of θ. θ^ is said to be unbiased estimate if E( θ^) = θ

2. Consistency

An estimate θ^ is said to be a consistent estimate of parameter θ if the probability that θ^ becomes closer and closer to θ approaches unity with an increase of sample size.

Another word: an estimate θ^ is said to be consistent whose variance decreases with the increase of sample size. Let θ^ be an estimate of θ based on a sample of size n. then θ^ is said to be a consistent estimate of θ.

3. Efficiency

An unbiased estimate is said to be efficient if the variance of its sampling distribution is smaller than that of the sampling distribution of any other unbiased estimate of the same parameter.

4. Sufficiency

An estimate is called a sufficient estimate. If the statistic used as an estimator uses all the information contained in the sample.

For example, the sample mean is based on all the observations of the sample, while the median and mode are based on a few observations; thus, the sample mean is a sufficient estimate. Practical criteria for sufficiency.

Confidence Interval Estimate

The point estimate is not a suitable estimate because the point estimate varies from sample to sample. This variation is overcome by the interval estimate by adding and subtracting the marginal error from the estimate.

Let θ^ be the estimate of θ, then the interval is obtained as:

Thus, a confidence interval estimate for parameter “ θ" is computed from the estimate "θ^" with a statement how confident the interval contains the unknown parameter.

Mathematically, it can be written as:

Where:

ME is dependent on the standard error and the statistical significance level (such as z, t, F, etc.).

Confidence Interval Estimate of the Mean of Normal Population

To construct a confidence interval for the mean "μ" of a normal population. The following two cases are to be considered.

Case 1: Confidence interval for the mean "μ" of a normal population when the standard deviation is known.

Let X¯ be the unbiased estimate of μ computed from the values of a random sample of size "n" selected from a normal population having mean "μ" and known standard deviation "σ.". The sampling distribution of sample mean approaches normal distribution with mean "μ" and standard deviation σ/sqrt(n). Whether the sample size is small or large.

The statistic is given by

Now to construct the (1-α)% confidence interval for μ is given by:

Case 2: Let X¯ be the unbiased estimate of μ computed from the values of a random sample of size "n" selected from a normal population having mean "μ" and unknown standard deviation "σ.". The sampling distribution of sample mean approaches normal distribution with mean "μ" and standard deviation S/sqrt(n). When the sample size is large.

The statistic is given by

Now to construct the (1-α)% confidence interval for μ is given by:

Example 7.4: A random sample consists of 4, 6, 8, 10, and 10 selected from a normal population having a mean "μ" and a standard deviation of σ = 3. Find 95% confidence interval for population mean "μ.".

Solution: As n = 5 and σ = 3.

Compute the sample mean:

1- α = 0.95

α = 0.05

95% confidence interval for population mean "μ." is given by

Hence 95% chances that the population mean μ lies between 4.93 and 10.23.

Example 7.5: A random sample of size n = 49 selected from a normal population yields the sample mean and standard deviation of 100 and 3.50, respectively. Find a 98% confidence interval for the normal population mean.

Solution:

n = 49, X−bar = 100, and σ = 3.50.

1- α = 0.98

α = 0.02

98% confidence interval for population mean "μ." is given by

Hence 98% chances that the population mean μ lies between 98.835 and 101.165.

Example 7.6: The summary information of a random sample of size 50 selected from a normal population having mean "μ" and unknown standard deviation of σ is given below:

Find a 95% confidence interval utilizing the above information.

Solution: The sample size n = 50 is large, and the sampling distribution of the sample mean is followed by z.

Now to calculate the sample mean and sample standard deviation.

95% confidence interval for population mean "μ." is given by

Hence 98% chances that the population mean μ lies between 2.277 and 2.723.

Example 7.7: A physician might think that a novel medication can lower a patient's blood pressure. He might enlist 20 individuals from a clinical record whose standard deviation is 4.30 to take part in a trial where they would take the novel medication for a month in order to test this. The physician may note that the average blood pressure is 140 in the sample at the end of the month. Find 95% confidence limits for overall patients listed in the clinic record.

Solution: As σ = 4.30 is known, the sampling distribution of average blood pressure follows z even n = 20