Statistics and Probability LAS Quarter 3 Week 4

 

Statistics and Probability

LAS Quarter 3 Week 3LEARNING ACTIVITY SHEET

QUARTER ___3_____/ SEMESTER ___2______

Week 4

Developed by: RICHARD R. RAQUEÑO

I. Title: Normal Distribution
II. Type of Activity: Concept notes with formative activities

III. MELC:
1. illustrates a normal random variable and its characteristics. M11/12SP-IIIc-1
2. identifies regions under the normal curve corresponding to different standard normal values. M11/12SP-IIIc-3
3. converts a normal random variable to a standard normal variable and vice versa.M11/12SP-IIIc-4
4. computes probabilities and percentiles using the standard normal table.M11/12SP-IIIc-d-1

 
IV. Learning Objective/s:
1. define a normal random variable
2. illustrate a normal random variable and its characteristics; and
3. state and apply the concepts of the empirical rule or 68%-95%-99.7% of the distribution.
4. identify the regions under the normal curve of different standard normal values.
5. use the z Table (Table of Areas under the Normal Curve) to find the regions that correspond to z values
6. sketch the normal curve showing the required regions or areas.
7. distinguish a raw score (x) and the standard score (z).
8. express normal random variable (x) as standard normal variable (z).
9. express standard normal variable (z) as normal random variable (x).
10. apply the concepts of normal random and standard normal variables in solving real-life problems.
11. utilize the use of the Table of Standard Normal Distribution
12. compute probabilities and percentiles related to a given z-score or normal random variable x; and
13. sketch the graph of the normal distribution.

VI. Concept notes with formative activities

Illustrating the Normal Random Variable and its Characteristics

Normal Probability Distribution is a probability distribution of continuous random variables. It shows graphical representations of random variables obtained through measurement like the height and weight of the students, the percentile ranks of the A&E (Accreditation and Equivalency) Test result of the Alternative Learning System students, or any data with infinite values.

It is used to describe the characteristics of populations and help us visualize the inferences we make about the population. It also used to determine the probabilities and percentile of the continuous random variables in the distribution. For example, your grades in Mathematics is one of the scores in the distribution, you can predict the location of that score in the distribution and interpret it with regards to the mean and standard deviation.

Properties of Normal Curve

The graphical representation of the normal distribution is popularly known as a normal curve. The normal curve is described clearly by the following characteristics:

The graphical representation of the normal distribution is popularly known as a normal curve. The normal curve is described clearly by the following characteristics:

1. The normal curve is bell-shaped.
2. The curve is symmetrical about its center. This means that, if we draw a segment from the peak of the curve down to the horizontal axis, the segment divides the normal curve into two equal parts or areas.
3. The mean, median, and mode coincide at the center. This also means that in a normal distribution, or a distribution described by a normal curve, the mean, median, and mode are equal.

4. The width of the curve is determined by the standard deviation of the distribution.
5. The tails of the curve are plotted in both directions and flatten out indefinitely along the horizontal axis. The tails are thus asymptotic to the baseline. A portion of the graph that is asymptotic to a reference axis or another graph is called an asymptote, always approaching another but never touching it.
6. The total area under a normal curve is 1. This means that the normal curve represents the probability, or the proportion, or the percentage associated with specific sets of measurement values.

A normally distributed random variable with a mean µ = 0 and standard deviation ơ = 1 is called a standard normal variable. It is presented using standard normal distribution where the center of the curve is zero, which is mean and added one unit from the center to the right and subtract one unit from the center to the left.

Standard Normal Distribution

 The shape of a normal curve is based on the two given parameters, the mean and the standard deviation of the distribution. When comparing two distributions each described by the normal curve, the following are the three situations based on the said parameters

 

a.      When the means are not equal, but the standard deviations are equal. (µ₁ ≠ µ_2; ơ₁=ơ­₂), the curves have a similar shape but centered at different points, as shown below:

a.      When the means are equal, but the standard deviations are equal (µ₁ = µ_{2 ;} ơ₁ ≠ ơ­₂ ), the curves are centered at the same point but they have different height and spreads. as shown in figure 5.

 

The means are equal, but the standard deviations are not equal.

 

b.     When the means are different, and the standard deviations are also different

(µ₁≠ µ_2; ơ₁ ≠ ơ­₂), the curves are centered at different points and vary in shapes, as shown in the figure below:

             

 The means are different, and the standard deviations are also different.

 c. When the means are different, and the standard deviations are also different (µ1≠ µ2; ơ1 ≠ ơ­2), the curves are centered at different points and vary in shapes, as shown in the figure below:

EMPIRICAL RULE

Empirical rule is used to roughly test the distribution’s normality, if many data of a random variable fall outside the lower and upper limits of the three-standard deviation, this means that the distribution is not normal

The empirical rule is better known as 68% - 95% - 99.70% rule. This rule states that the data in the distribution lies within one (1), two (2), and three (3) of the standard deviation from the mean are approximately 68%, 95%, and 99.70%, respectively. Since the area of a normal curve is equal to 1 or 100% as stated on its characteristics, there are only a few data which is 0.30% falls outside the 3-standard deviation from the mean.

 

For instance, the distribution of the grades of the Senior High School students in Statistics and Probability for the Third Quarter is shown below.

Using the empirical rule, you can say that the distribution of the grades of the Senior High School in Statistics and Probability are distributed normally because the shape of the graph is bell-shaped and symmetric about its mean, which is 87. This also indicates that most of the students’ grades fall close to the mean and fewer of them are far below and/or above the mean

It can summarize the distribution in the following percentage:

• 68% of data lies within 1 standard deviation from the mean have a grade of 83 to 91
• 95% of data lies within 2 standard deviations from the mean have a grade of 79 to 95
• 99.70% of data lies within 3 standard deviations from the mean have a grade of 75 to 99

Example 1:   

The scores of the Senior High School students in their Statistics and Probability quarterly examination are normally distributed with a mean of 35 and a standard deviation of 5.

Answer the following questions:
a. What percent of the scores are between 30 to 40?
b. What scores fall within 95% of the distribution?

Solution:

Draw a standard normal curve and plot the mean at the center. Then, add the standard deviation to the mean once and mark it to the right of the mean. Add twice the standard deviation to the mean and put it to the right of the first sum. Then, add thrice the standard deviation to the mean and mark it to the right of the second sum. Do the same to the left. This time, instead of adding, subtract the standard deviation from the mean.

Distribution of Scores of Senior High School Students

Answer:

a. The scores 30 to 40 falls within the first standard deviation from the mean. Therefore, the scores that fall between 30 and 40 is approximately 68% of the distribution.

b. Since 95% of the distribution lies within 2 standard deviations from the mean, then the scores corresponding to this area of the distribution are scores from 25 up to 45.

Example 2:

The school nurse of Ilocos Sur NHS needs to measure the BMI (Body Mass Index) of the Alternative Learning System students. She found out that the heights of male students are normally distributed with a mean of 160 cm and a standard deviation of 7 cm. Find the percentage of male students whose height is within 153 cm to 174 cm.               

Distribution of BMI of the Alternative Learning System Students

Solution:

The mean of this problem is 160, it is subtracted by 7 to the left (e.g., 160 -7 = 153; 153-7 = 146; 146 – 7 = 139) and add 7 to the right. (e.g., 160 + 7 = 167; 167 + 7 = 174 + 7 = 181).

 As stated in figure 8, 153 cm falls at 1 standard deviation from the mean to the left and the height of 174 cm falls at 2 standard deviations from the mean to the right.  Therefore, it covers the whole 68% and 13.5%. of the distribution and the sum of it is 81.5%

Answer: 81.5% of the male students have a height between 153 cm to 174 cm.

BELOW IS THE FULL COPY OF THIS LAS