Areas Under the Normal Distribution Curve

Figure 1. Normal Distribution Table

Finding Areas Under the Normal Distribution Curve

          In finding areas under the normal curve, used table in Figure 1. This table provides areas under the standard normal curve below a specified value z. The first column in table of normal curve refers to the values of the standard score or z-score ranging from 0.0 to +3.0 standard deviations from the mean. The first-row heading specifies the value of z-score in two decimal places while the entries in the table are the areas below a given value of z, in four-decimal-place accuracy.

Example 1: Determine the area under the standard normal curve of z = 2.49.

Solution: First go down the left-hand column, labeled z to “2.4”. Then go cross that row until the column 0.09. That intersection is the area under the standard normal curve of z = 2.49, that is, 0.4936.

Example 2: Determine the area under the standard normal curve of z = -1.38.

Solution: Apply the property of the normal curve that is, the normal curve is symmetrical about the mean. Thus, the area under the normal curve of z = -1.38 is just the same in finding areas under the normal curve of z = 1.38. First go down the left-hand column, labeled z to “1.3”. Then go cross that row until the column 0.08. The intersection is the area under the standard normal curve of z = -1.38, that is, 0.4162.

Example 3: Find the area under the standard normal curve to the right of z = 0.25.

Solution: First go down the left-hand column, labeled z to “0.2”. Then go cross that row until the column 0.05. That intersection is the area under the standard normal curve from 0 to z = 0.25 which is 0.0987, thus, the area under the standard normal curve to the right of z = 0.25 is the difference between 0.5 and obtained area or 0.5 – 0.0987 = 0.4013.

Example 4: Find the area under the standard normal curve to the left of z = 1.73.

Solution: First go down the left-hand column, labeled z to “1.7”. Then go cross that row until the column 0.03. That intersection is the area under the standard normal curve from 0 to z = 1.73 which is 0.4582, thus, the area under the standard normal curve to the left of z = 1.73 is the sum between 0.5 and obtained area or 0.5 + 0.4582 = 0.9582.

Example 5: Find the area under the standard normal curve to the left of z = -1.08.

Solution: First go down the left-hand column, labeled z to “1.0”. Then go cross that row until the column 0.08. That intersection is the area under the standard normal curve from 0 to z = -1.08 which is 0.3599, thus, the area under the standard normal curve to the left of z = -1.08 is the difference between 0.5 and obtained area or 0.5 – 0.3599 = 0.1401.

Example 6: Find the area under the standard normal curve above z = -2.46.

Solution: First go down the left-hand column, labeled z to “2.4”. Then go cross that row until the column 0.06. That intersection is the area under the standard normal curve from 0 to z = -2.46 which is 0.4931, thus, the area under the standard normal curve to the left of z = -2.46 is the sum between 0.5 and obtained area or 0.5 + 0.4931 = 0.9931.

Example 7: Find the area under the standard normal curve between z = 0.82 and z = 2.72.

Solution: The area between 0 and 0.82 is 0.2939 and the area between 0 and 2.72 is 0.4967. Thus, the required area between the z values 0.82 and 2.72 is the difference between these two areas or 0.4967 – 0.2939 = 0.2028.

Example 8: Determine the area under the standard normal curve between z = -1.57 and z = 1.63.

Solution: The area between -1.57 and 0 is 0.4418 and the area between 0 and 1.63 is 0.4484. Thus, the required area between the z values -1.57 and 1.63 is the sum between these two areas or 0.4484 – 0.4418 = 0.8902.

Standardizing a Normal Curve

          Most of the time, the probabilities you are interested in will involve a random variable let say x, a normal random variable with mean μ and standard deviation σ. You must then standardize variable of interest, writing it as the equivalent form in terms of z-score the standard normal random variable. Once this is done, the probability of interest is the area that you find using the standard normal probability distribution. The formula to convert raw score into standard score is:

Example 1: For a continuous random variable that has a normal distribution with a mean of 20 and a standard deviation of 4, find the area under the normal curve from 19 and 26.

Now, the area between -0.25 and 0 is 0.0987 and the area between 0 and 1.5 is 0.3531. Therefore, the area under the normal curve between 19 and 26 is sum between these two areas or 0.3531 + 0.0987 = 0.4518.

Example 3: A brisk walk at 4 miles per hour burns an average of 300 calories per hour. If the standard deviation of the distribution is 8 calories, find the probability that a person who walks one hour at the rate of 4 miles per hour will burn the following calories. Assume the variable to be randomly normally distributed.

a. more than 280 calories.

b. less than 294 calories.

c. between 278 and 318 calories.

The area from -0.75 and 0 is 0.2734. Thus, the probability that a person who walks one hour at the rate of 4 miles per hour will burn less than 294 calories is the difference between 0.5 and the area or 0.5 – 0.2734 = 0.2266. Therefore, there is a 22.66% probability that a person who walks one hour at the rate of 4 miles per hour will burn less than 294 calories.

The area between -2.75 and 0 is 0.4970 and the area between 0 and 2.25 is 0.4878., the probability is the sum between these two areas or 0.4878 + 0.4970 = 0.9848. Therefore, around 98.48% probability that a person who walks one hour at the rate of 4 miles per hour will burn between 278 and 318 calories respectively.

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SAMPLE SPACE AND EVENTS OF AN EXPERIMENT INTRODUCTION to PROBABILITY

Estimation of Parameters

Learning Objectives:

Given the learning materials and activities of this chapter, they will be able to:

Ø  Distinguish classical methods from Bayesian methods of estimates.

Ø  Calculate the standard error of a sample.

Ø  Calculate the margin of error for interval estimate.

Ø  Construct interval estimates of the population mean given a specified level of confidence.

Ø  Construct interval estimates of the population proportion with the specified confidence level.

Introduction

         There are two major areas in statistical inference, the estimation of parameters and hypothesis testing. Estimation is the process of estimating the value of a parameter from information obtained in a sample. An important aspect of estimation is the size of the sample. An estimator is a formula, the function or procedure used in estimating a population parameter. There are two methods of estimating population parameters, such as:

a.                       Classical method is based strictly on information obtained from a random sample selected from the population.

b.                 Bayesian method utilizes prior subjective knowledge about the probability distribution of the unknown parameters in conjunction with the information provided from the sample data.

In this text, we shall utilize the classical method to estimate unknown population parameters such as the mean, proportion and the variance by computing statistics from a random sample and applying the theory of sampling distributions. There are two ways in the classical method of estimation, namely: point estimate and interval estimate.

Point estimate

            Point estimate consists of a single value used to estimate a population parameter. For most parts, the point estimate will be different from the population mean due to sampling error. There is now way of knowing how close the point estimate is to the population parameter. For this reason, statisticians prefer another type of estimate.

Interval estimate

            Is an interval or a range of values used to estimate the parameter. In an interval estimate, the parameter is specified as being between two values. A degree of confidence can be assigned before an interval estimate is made. The confidence level is the probability that the interval estimate will contain the true population mean or population proportion.

        Three common confidence level are 90%, 95% and 99% confidence intervals. The table below summarizes the values of the standard deviates and the margin of error for the most commonly used confidence level.

       A term level of significance is defined as the probability of erroneously concluding that a confidence interval generated will contain the parameter. As observed in the table, the greater the level of confidence, the larger the z values, the larger the margin of error and of course the wider the confidence interval.

        An interval estimate is constructed by subtracting and adding the margin of error to a point estimator. The length of the confidence interval is determined by the sample size, the standard deviation, and the desired confidence level.

Estimating Means

         Estimation (or estimating) is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available. Typically, estimation involves "using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter". The sample provides information that can be projected, through various formal or informal processes, to determine a range most likely to describe the missing information. An estimate that turns out to be incorrect will be an overestimate if the estimate exceeded the actual result, and an underestimate if the estimate fell short of the actual result. 

Estimating Means Large Sample and the Standard Deviation is known

The Central Limit Theorem says that, for large samples (samples of size n ≥ 30), when viewed as a random variable the sample mean is normally distributed with mean and standard deviation. The Empirical Rule says that we must go about two standard deviations from the mean to capture 95% of the values of sample mean generated by sample after sample.

Confidence interval for means >=30 and the standard deviation is known the formula is

Example 1: A study of 40 bowlers showed that their average score was 186. The standard deviation of the population is 6.

a.       Find the 95% confidence interval of the mean score for all bowlers.

b.      Find the 99% confidence interval of the mean score of a sample of 100 bowlers instead of a sample of 40.

          Thus, it can be 95% confident that the true mean score of bowlers is between 184.14 and 187.86. This means that 95% of the time, the population mean score of bowlers will be roughly between 184 and 188. 

          Thus, the 99% confidence interval for the population mean score is ranging from 184.542 to 187.548. This means that we can be 99% confident that the population mean score is roughly between 185 to 188.

Confidence interval for means < 30 and the standard deviation is unknown (small sample)

         When the population standard deviation is unknown and the sample size is less than 30, the standard deviation from the sample can be used in place of the population standard deviation. In this case, the t-distribution is used to determine the confidence interval and the random variable is approximately normally distributed. The formula is:

          To determine the value of t critical locates the critical value from the table in t distribution with the corresponding degrees of freedom. The degrees of freedom are the values that are free to vary after a sample statistic has been computed. The degrees of freedom for the confidence interval for the mean is n – 1. Also, note that the sample standard deviation is used instead of the population standard deviation. 

Example 2: A sample of 20 tuna showed that they swim an average of 8.6 miles per hour. The standard deviation for the sample was 1.6. Find the 95% confidence interval of the true mean.

Solution: Given n = 20, the sample mean of 8.6 miles per hour, and the sample standard deviation s = 1.6 miles per hour. The degrees of freedom is n – 1 = 20 – 1, using the t-distribution table yielded a critical value of t =2.093. Hence, 

          Thus, the 95% confidence interval for the population mean time is ranging from 7.851 to 9.349 miles per hour. This means that we can be 95% confident that the true mean time of tuna can swim roughly between 8 and 9 miles per hour. 

When to use the z and t distribution:

-          If the population standard deviation is known and sample size is large, use z-test.

-          If the population standard deviation is unknown and sample size is large, use z-test.

-          If the population standard deviation is unknown and sample size is small, use t-test.

Estimating Proportions

       When the variable of interest is qualitative and are summarized in terms of frequencies, confidence intervals for estimating proportion may be constructed. 

          To construct confidence interval for estimating a population proportion based on a proportion obtain from a random sample, similar procedure used to estimate population mean. 

          Example 3: A local polling organization reports that based on a local-wide survey of 500 respondents, 43% of the vote will be in favor of the administration governatorial candidate in the May 2016 elections. Construct the 95% confidence interval for the proportion indicating preference for the administration candidate.

         Thus, the 95% confidence interval is from 38.7% to 47.3%. This means that with a sample of 500, the poll has a margin of error of ±4.3% and the pollster can be 95% confident that the administration candidate will obtain roughly between 39% and 47% of the votes.

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Population and Sample, Parameter and Statistic, Descriptive and Inferential

What is Population? Sample?

In statistics, researchers, and educators commonly use the terms population and sample (Alferez & Duro, 2006). The population is often too large for us to examine each of its members.

           The population is the entire collection of all elements/experimental units under consideration in a statistical inquiry or to be studied. Sample a part of the population. If the sample is to provide information about the entire population, it must be representative of that group in some way. In actuality, unless a sample is picked at random, it cannot be expected to be representative of a population. This is because any nonrandom rule for selecting a sample will almost always produce one that is skewed toward some data values over others.

        For example, if we wish to determine the average income of households in Zamboanga del Norte, then the population of interest is the collection of all households in Zamboanga del Norte. However due to some constraints, the budget, the time, the manpower for instance, and then we would have to redefine the interest. This time we can delimit the scope of the study we utilized sampling size to include only the collection of all households in Dapitan and Dipolog City, considering this is done thru sampling techniques.

What is Parameter? Statistic?
          Parameter refers to any numerical value describing a characteristic of a population and is usually denoted by some Greek letters such as population standard deviation, σ and population mean, μ. 

         While the term statistic is a numerical measurement describing some characteristics of a sample. The symbols x and s are statistics which is unbiased of the parameter μ and σ.

Two Major Areas of Statistics

Descriptive Statistics

Is concerned with the methods for collecting, organizing, and describing a set of data so as to yield meaningful information (Walpole, 2000). Construction of tables, charts, and graphs and the computation of descriptive statistical measures also fall in this area.

Inferential Statistics

Inferential Statistics is also called Inductive Statistics or Statistical Inference. Comprises those procedures for drawing inferences or making generalizations about characteristics of a population-based on partial and incomplete information obtained sample data to infer to populations.

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Introduction to Statistics