Inferential Statistics Notes

Inferential Statistics

Statistical inference is a process by which we infer population properties from sample properties.
It's a method of making informed guesses or decisions about a large group (population) based on a smaller subset (sample).

Aims to guess, predict, or conclude the true value of a population parameter using a sample.
Involves selecting a random sample from a population and using a sample statistic to estimate a population parameter.
For example, the sample mean (\bar{X}) is used to estimate the population mean \mu .

Point Estimation
- Draws inference about estimating the value of an unknown parameter using a single value or point.
Interval Estimation
- Draws inference about estimating the value of an unknown parameter using a range of values with high probability.

Draws inference about estimating the value of an unknown parameter using a value or a point.

If there are many estimators of the same population parameter, choose the estimator whose sampling distribution has the smaller variance.

A student wants to know the average grade of the students in Statistics and Probability and interviews 40 students to estimate it.
Ana owns a small store and wants to estimate the average amount her customers spend per day. She takes a sample of 10 customers and records their spending: ₱120, ₱150, ₱180, ₱140, ₱130, ₱160, ₱175, ₱155, ₱145, and ₱170. Using point estimation, the estimated average amount a customer spends in her store can be calculated.
The monthly salary of Call Center Agents is reflected below. Their monthly average income can be estimated using point estimation. Then, if Allan earns Php32,250.00, it can be determined whether he is included in the average income bracket.
In the year 2015, the municipal registrar reported that the average matrimonial age for male person is 26.8 years old. Date on April 6, 2016 showed the age of 5 male persons getting married are 24, 28, 21, 31, and 27 while on May 3, 2016 the age of 8 male persons getting married are 20, 33, 30, 28, 35, 21, 27 and 24. Determine the (a) unbiased estimator and (b) most efficient estimator.
Mark is a teacher who wants to estimate the average score of his students in a recent math exam. He randomly selects 2 sets of 10 students and records their scores:
- SET A: 78, 85, 92, 88, 76, 80, 95, 90, 84, 79
- SET B: 98, 78, 89, 94, 91, 88, 82, 77, 84, 82
- Using an unbiased and efficient estimator, the best estimate for the true average score of all students can be determined.

Draws inference about estimating the value of an unknown parameter using a range of values with high probability.
Involves a certain level of confidence.
Confidence Level - refers to the percentage of all possible samples that can be expected to include the true population parameter.

Standard Error - a measure of the variability of a statistic. It is an estimate of the standard deviation of a sampling distribution.
Margin of Error – expresses the maximum expected difference between the true population parameter and a sample estimate of that parameter.

To determine the average weight of oranges, 100 oranges were weighed. The mean weight of the sample is 150 g. Previous studies show a standard deviation of 40 g. The interval estimate of the population mean μ using a 95% confidence level can be found.
In a study of 50 senior high school students, the mean number of hours per week that they played video games was 20.5 hours. It was estimated that the population standard deviation was 3.7. The 99% confidence interval for the mean time for playing video games can be calculated.
A factory produces light bulbs, and the company wants to estimate the average lifespan of its bulbs. A random sample of 40 bulbs is tested, and the sample mean lifespan is found to be 1,200 hours. The population standard deviation is known to be 100 hours. A 95% confidence interval for the true mean lifespan of the light bulbs can be constructed.

A random sample of 30 students were measured about their height. The average height of the sample is 5.3 feet. The sample standard deviation is 1.1 feet. The interval estimate of the population mean μ using 95% confidence level can be found.
Rochelle wants to know the mean of all entering trainees in a boot camp. The mean age of a random sample of 25 trainees is 18 years and the standard deviation is 1.3 years. The sample comes from a normally distributed population. The interval estimate of the population mean μ using 99% confidence level can be found.
A researcher wants to estimate the average time high school students spend on homework each week. She randomly selects 25 students and records their weekly study hours. The sample mean is 12.5 hours, and the sample standard deviation is 3.2 hours. A confidence interval can be constructed to estimate the population mean.
A nutritionist wants to estimate the average daily protein intake (in grams) of adult women in a city. She randomly selects 16 women and records their daily protein intake. The sample mean is 55 grams, and the sample standard deviation is 8 grams. A 99% confidence interval for the population mean daily protein intake can be constructed; assume the data is approximately normally distributed.