Fundamental Sampling Distributions: A Comprehensive Overview, Slides of Data Analysis & Statistical Methods

Download Fundamental Sampling Distributions: A Comprehensive Overview and more Slides Data Analysis & Statistical Methods in PDF only on Docsity!

7: Fundamental

Sampling Distributions

EM 7: Engineering Data Analysis Second Semester, 2019- Pamantasan ng Lungsod ng Valenzuela

Random Sampling

2

Population and Samples

For example, if one makes a random selection of n = 8 storage batteries from a manufacturing process that has maintained the same specification throughout and records the length of life for each battery, with the first measurement 𝑥ଵ being a value of 𝑋ଵ, the second measurement 𝑥ଶ being a value of 𝑋ଶ, and so forth, then 𝑥ଵ, 𝑥ଶ, … , 𝑥଼ are the values of the random sample 𝑋ଵ, 𝑋ଶ, … , 𝑋଼. (^4) DEFINITION: Let 𝑋ଵ, 𝑋ଶ, … , 𝑋௡ be n independent random variables, each having the same probability distribution 𝑓(𝑥). Define 𝑋ଵ, 𝑋ଶ, … , 𝑋௡ to be a random sample of size n from the population 𝑓(𝑥) and write its joint probability distribution as, 𝑓 𝑥ଵ, 𝑥ଶ, … , 𝑥௡ = 𝑓 𝑥ଵ 𝑓 𝑥ଶ … 𝑓(𝑥௡)

Statistic

• Oftentimes, it is impossible to obtain a value for a parameter p representing the population proportion.
• Instead, a large random sample is selected and the proportion 𝑝̂ of the population where it is calculated.
• Now 𝑝̂ is a function of the observed values in the random sample. Since many random samples are possible from the same population. 5

Location Measures of a

Sample

• Sample mean: 𝑋ത^ =

௡ ௜ୀଵ

• Sample median: 𝑥෤ = ൞

ଶ

ଶାଵ^

• Sample mode is the value of the sample that occurs most often 7

Example

Suppose a data set consists of the following observations: 0.32 0.53 0.28 0.37 0.47 0.43 0. 0.42 0.38 0. Calculate mean, median and mode 8

Variability Measures of a

Sample

• Sample variance: 𝑆ଶ^ =

෍ 𝑋௜ − 𝑋ത^ ଶ

௡ ௜ୀଵ The computed value of 𝑆ଶ^ for a given sample is denoted by 𝑠ଶ. 10

Example

A comparison of coffee prices at 4 randomly selected grocery stores in Pasig showed increases from the previous month of 12, 15, 17, and 20 pesos for a 1-kg bag. Find the variance of this random sample of price increases. 11 X-bar = 16 pesos S^2 = 34/

Example

Find the variance of the data, 3, 4, 5, 6, 6, and 7, representing the number of tilapia caught by a random sample of 6 fishermen on October 7, 2019, at Laguna de Bay. 13 Sum x^2 from 1 to 6 = 171 Sum x, from 1 to 6 = 31 n = 6 s^2 = 13/ Std = 1. Sample range = 7-3 = 4

Sampling Distributions

• The field of statistical inference is basically concerned with generalizations and predictions. For example, we might claim, based on the opinions of several people interviewed on the street, that in a forthcoming election, 60% of the eligible voters in Marikina favor a certain candidate. (Random sample of opinions from a very large finite population).
• 2nd: The average cost to build a residence in Cagayan, is between Php 330,000 and Php 335,000, based on the estimates of 3 contractors selected at random from the 30 now building in the province. 14

Sampling Distributions

• The probability distribution of a statistic is called a sampling distribution.
• The sampling distribution of a statistic depends on the distribution of the population, the size of the samples, and the method of choosing the samples.
• The probability distribution of 𝑋ത^ is called the sampling distribution of the mean. 16

Sampling Distribution of

Means

Suppose that a random sample of n observations is taken from a normal population with mean 𝜇 and variance 𝜎ଶ: 𝑋ത^ = 1 𝑛 𝑋ଵ + 𝑋ଶ + ⋯ + 𝑋௡ has a normal distribution with mean, 𝜇௑ത = 1 𝑛

• 𝜇 + 𝜇 + ⋯ + 𝜇 = 𝜇 and variance, 𝜎௑^ ଶത^ = 1 𝑛ଶ^ 𝜎ଶ^ + 𝜎ଶ^ + ⋯ + 𝜎ଶ^ = 𝜎ଶ 𝑛 If sampling from a population with unknown distribution, the sampling distribution of 𝑋ത^ will still be approximately normal with mean 𝜇 and variance 𝜎ଶ/𝑛 provided that the sample is still large. 17

19

Example

A cement manufacturing company manufactures cements that have a shelf life that is approximately normally distributed, with mean equal to 800 days and a standard deviation of 40 days. Find the probability that a random sample of 16 cement bags will have an average life of less than 775 days. 20