## Year 10-12 Learning Continuity Packages

### 29 May, 2020: Probability 2 Student Learning Continuity Program
29 May, 2020: Probability 2 /

This is the second Mathematics – Probability session for the Student Learning Continuity Program by MEHRD.

### Focus: Statistics Minor Project

#### Quantitative/Qualitative data

Quantitative data– discrete and continuous variable

Variable– Characteristic under investigation that assumes different values to different elements, eg Families income,  heights of students, age of students, number of words in a page of a book etc

#### Types of variables:

Discrete variable – A variable whose values are countable. Assume only certain values, but not intermediate values

Frequency distribution (Ungrouped)

Frequency distribution table, Mean, Median, Mode, Standard deviation, Bar graph, Cumulative frequency distribution, Lower quartile (Q1), Second quartile (Q2) or median, Upper Quartile (Q3)

Continuous Variable – A variable that can only assume any numerical value over a certain interval…eg length of longest finger, Height and weight of students.

Frequency distribution (Grouped)

Number of classes, Class boundary, Class width, Class mid-point, Modal class, median class, Mean, Cumulative frequency distribution table and graph (Q1, Q2, Q3) .

Random sampling

Sampling method- a kind of so that every member within the population to have equal chances to be selected in the sample

Why sampling survey but not a census?

• Reduce cost
• Census is time consuming
• Sometimes it is impossible to conduct a census. It is sometimes difficult to access every member of the population

Element– Each member/item of the population being investigated.

Raw data: Data recorded in the sequence in which they are collected and before they are processed or ranked.

Go to your library and read about how to estimate a population mean by using random selected sample.

• Consider why we do sampling
• Is the sampling method not bias
• How large is my sample size
• Define population parameters and sample statistics

Hypothesis

In formulating a hypothesis, students need to make an educated guess. For example “The mean age of Forms 4 and 5 students at Kombito Community High School is 16 years”.

Aim: For students to produce a written report about their statistical investigation of a randomly collected sample of size 30 numerical items from a population of approximately 200.

Objectives

• To know how to formulate a hypothesis
• To know how to draw a random sample of certain size from a population
• To know how to treat the data such as arranging data in frequency distribution table, draw types of graphs
• To know how to find the central tendency
• To estimate the population parameters by using the sample statistics with the help of 95% CI

Chapter 1: The Topic

• Identify a population and what variable of the population you are going to investigate.
• Sampling method and outline how you will go about selecting your sample of size 30. Generate 40 random numbers
• Hand in Chapter 1

Chapter 2: Sampling and Data collection

•  Outline the sample selection process
• Tabulate the sample being collected as your raw data

Chapter 3: Data presentation and analysis

•  Display data using appropriate frequency distribution tables
• Use not more than two types of graphs to display data
• Calculate the measures of central tendency (mean, median, mode)and what they tell you about the data
• Calculate the measures of the spread (Range, quartiles, standard deviation) of the distribution and what they tell you about the data
• Calculate the 95% confidence interval and what is its significance

Chapter 4: Findings and conclusion

•  A well summary of findings
• Discuss the 95% confidence interval
• Conclusion relating the findings to the hypothesis

Calculating the 95% confidential interval Note that is the standard deviation of the sample mean. As . Thus as n increases sample mean is approximately equal to the population mean.

The Normal Distribution

Frequency distributions can assume almost any shape or form, depending on the data. However, the data obtained from many experiments often followed a common pattern. For example, heights of students, weights of student and students’ exam scores all lead to data that have the same kind of frequency distribution. This distribution is referred to as the normal distribution. Because it occurs so often in practical situations more either exactly or approximately. It is generally regarded as the most important distribution and much statistical theory is based on it.

The normal probability distribution is a bell-shaped curve. Its mean is denoted by µ and its standard deviation by σ. A normal probability distribution when plotted gives a bell-shaped curve such that

 The total area under the curve is 1 or 100% The curve is symmetric about the mean. 50% of the total area under a normal distribution curve lies left side of the mean and another 50% lies right side of the mean. The two tails of the curve extend indefinitely.

Of the area under the curve, approximately:

1. 68% lies within one standard deviation from the mean. (We can say “likely”).
2. 95% lies within two standard deviations from the mean. (We can say “very likely”).
3. 99% lies within three standard deviations from the mean. (We can say “almost certain”).

Exercise

1. X has a normal distribution with a mean of 6 and a standard deviation of 2 Find the following percentages.
1. The percentage between 4 and 8.
2. The percentage between 4 and 12.
3. The percentage less than 2.

1. X has a normal distribution with a mean of 30 and a standard deviation of 5 units.
1. Between what two values is X likely to be?
2. Above what value is X very likely to be?
3. X is almost certain to be less than what value?

NOTE:

Three (3) types of mean that will be looked at in the minor research project.

1. Estimated mean —- This is the mean that was stated in the hypothesis question/statement.
2. Sample mean —- This is calculated mean form the 30 samples you randomly selected. Note that this mean is also representing the population mean because the samples elements are taken from the population.
3. Population mean—– This mean is the one that we are finding the limits (range) that it will fall in because it is time consuming if you are going to find the mean of a population of 200 or more. Therefore, find the 95% confidence interval.

Conclusion must capture the calculated values, graphs, tables and most importantly the hypothesis question.

Do something extra. This means you need suggest ways to prevent, stop or ways to deal with the situation. For example, if you found out that a lot of positive malaria cases in your school during the year. What steps should the students, school administration and surrounding communities do to reduce the malaria cases.

Download this information as a word document: Form 6 Maths Lesson 2

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