BPCC-104 Solved Assignment

1. Define statistics. Explain the basic concepts in statistics.
2. Compute Pearson’s Product Moment Correlation for the following data:

3. Explain the type-I and type-II errors in the process of hypothesis testing.
4. Compute mean, median and mode for the following data

5. Explain the merits, limitations and uses of standard deviation.
6. Compute average deviation for the following data.

7. Describe the properties of Normal Distribution Curve (NPC)
8. Discuss the concept of Kurtosis and Skewness.

Expert Answer:

Define statistics. Explain the basic concepts in statistics.

Answer:

Definition of Statistics:

Basic Concepts in Statistics:

1. Population and Sample:
   • Population: The entire group of individuals or items that we want to study. It includes every possible outcome or observation.
     • Example: All the students in a university.
   • Sample: A subset of the population selected for analysis. A sample is used to draw conclusions about the population without needing to study the entire population.
     • Example: A group of 200 students randomly selected from a university.
2. Descriptive and Inferential Statistics:
   • Descriptive Statistics: Involves summarizing and organizing data to describe its basic features. It includes measures such as mean, median, mode, variance, and standard deviation.
     • Example: Finding the average height of a group of people.
   • Inferential Statistics: Involves making predictions or inferences about a population based on sample data. It includes hypothesis testing, confidence intervals, and regression analysis.
     • Example: Predicting the average height of all students at a university based on a sample.
3. Variables:
   • Qualitative (Categorical) Variables: These describe characteristics that cannot be measured numerically but can be classified into categories. Examples include gender, nationality, or color.
   • Quantitative (Numerical) Variables: These describe measurable quantities and can be divided into two types:
     • Discrete Variables: Countable values, like the number of students in a class.
     • Continuous Variables: Any value within a given range, like height or temperature.
4. Measures of Central Tendency:
   These are metrics that summarize the central point of a data set:
   • Mean: The arithmetic average of a data set, found by summing all the values and dividing by the total number of values.
   • Median: The middle value in a sorted data set. If the data set has an even number of values, the median is the average of the two middle numbers.
   • Mode: The value(s) that occur most frequently in a data set.
5. Measures of Dispersion:
   These metrics describe how spread out the data is:
   • Range: The difference between the largest and smallest values in a data set.
   • Variance: The average of the squared differences between each value and the mean.
   • Standard Deviation: The square root of the variance, which provides a measure of the average distance from the mean.
6. Probability:
   Probability is a measure of the likelihood that a given event will occur. It ranges from 0 (impossible event) to 1 (certain event). It forms the foundation of inferential statistics, allowing statisticians to make predictions about populations based on sample data.
7. Normal Distribution:
   A normal distribution is a bell-shaped curve where most of the data points cluster around the mean, with fewer observations appearing as we move away from the mean. It is important in statistics because many natural phenomena follow this distribution, and it forms the basis for various statistical tests.
8. Hypothesis Testing:
   A statistical method used to decide whether there is enough evidence to support a particular belief (hypothesis) about a population. It involves:
   • Null Hypothesis (H₀): A default assumption that there is no effect or difference.
   • Alternative Hypothesis (H₁): The hypothesis that suggests there is an effect or difference.
   • Hypothesis testing aims to reject or fail to reject the null hypothesis based on sample data.
9. Correlation and Regression:
   • Correlation: Measures the strength and direction of a linear relationship between two variables. The correlation coefficient (r) ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).
   • Regression: A statistical technique for modeling the relationship between a dependent variable and one or more independent variables. It is used for predicting values and understanding relationships.

Summary:

Compute Pearson’s Product Moment Correlation for the following data:

Answer:

Step-by-Step Solution:

Formula for Pearson’s Correlation Coefficient:

• $A$$A$AA$A$ and $B$$B$BB$B$ are the data sets.
• $n$$n$nn$n$ is the number of data points.

Step 1: Compute the required summations:

• $\sum A$$\sum A$sum A\sum A$\sum A$: Sum of all values in Data A.
• $\sum B$$\sum B$sum B\sum B$\sum B$: Sum of all values in Data B.
• $\sum A^2$$\sum A^2$sumA^(2)\sum A^2$\sum A^2$: Sum of the squares of values in Data A.
• $\sum B^2$$\sum B^2$sumB^(2)\sum B^2$\sum B^2$: Sum of the squares of values in Data B.
• $\sum A\cdot B$$\sum A\cdot B$sum A*B\sum A \cdot B$\sum A\cdot B$: Sum of the product of corresponding values of A and B.

Data Table:

Summations:

Step 2: Apply the formula:

• $n=10$$n=10$n=10n = 10$n=10$
• $\sum A=190$$\sum A=190$sum A=190\sum A = 190$\sum A=190$
• $\sum B=234$$\sum B=234$sum B=234\sum B = 234$\sum B=234$
• $\sum A^2=4436$$\sum A^2=4436$sumA^(2)=4436\sum A^2 = 4436$\sum A^2=4436$
• $\sum B^2=5964$$\sum B^2=5964$sumB^(2)=5964\sum B^2 = 5964$\sum B^2=5964$
• $\sum A\cdot B=4235$$\sum A\cdot B=4235$sum A*B=4235\sum A \cdot B = 4235$\sum A\cdot B=4235$

Step 3: Simplify the numerator and denominator:

Numerator:

Denominator:

Step 4: Compute Pearson’s correlation:

Conclusion:

Explain the type-I and type-II errors in the process of hypothesis testing.

Answer:

Type I and Type II Errors in Hypothesis Testing

1. Type I Error (False Positive):

• A Type I error occurs when we reject the null hypothesis (H₀) when it is actually true.
• It is essentially a "false positive" result because it indicates that an effect or difference exists when, in reality, it does not.

Example:

• If the test concludes that the patient has the disease (rejecting H₀) when the patient is actually healthy (H₀ is true), a Type I error has occurred.

Probability of a Type I Error:

• The probability of committing a Type I error is denoted by $\alpha$$\alpha$alpha\alpha$\alpha$ (alpha), which is called the significance level of the test.
• Commonly, $\alpha$$\alpha$alpha\alpha$\alpha$ is set at 0.05 or 5%, meaning there’s a 5% risk of rejecting the null hypothesis when it is true.

Consequences:

• In practice, Type I errors can lead to false conclusions, such as claiming that a new drug is effective when it is not, or approving a faulty product for release.

2. Type II Error (False Negative):

• A Type II error occurs when we fail to reject the null hypothesis (H₀) when it is actually false.
• It is essentially a "false negative" result because it suggests that no effect or difference exists when, in reality, it does.

Example:

• If the test concludes that the patient is healthy (fails to reject H₀) when the patient actually has the disease (H₀ is false), a Type II error has occurred.

Probability of a Type II Error:

• The probability of committing a Type II error is denoted by $\beta$$\beta$beta\beta$\beta$ (beta).
• The power of the test (1 – $\beta$$\beta$beta\beta$\beta$) represents the ability of the test to detect an effect if one exists. A higher power reduces the likelihood of a Type II error.

Consequences:

• Type II errors can result in missed opportunities, such as failing to detect a beneficial effect of a new drug, or not identifying a problem in a manufacturing process.

Summary of Type I and Type II Errors:

Balancing Type I and Type II Errors:

• Reducing Type I Errors: Lowering $\alpha$$\alpha$alpha\alpha$\alpha$ (significance level) reduces the likelihood of Type I errors, but it increases the probability of Type II errors.
• Reducing Type II Errors: Increasing the sample size or conducting more sensitive tests can reduce $\beta$$\beta$beta\beta$\beta$, thereby lowering the chance of Type II errors, but it may increase the risk of a Type I error.

Compute mean, median and mode for the following data:

Answer:

1. Mean:

• Add all the numbers together.
• Divide the sum by the total count of numbers.

1. Sum of the data:
   $50+68+90+45+75+48+72+65+58+50=621$$50+68+90+45+75+48+72+65+58+50=621$50+68+90+45+75+48+72+65+58+50=62150 + 68 + 90 + 45 + 75 + 48 + 72 + 65 + 58 + 50 = 621$50+68+90+45+75+48+72+65+58+50=621$
2. Count of numbers: There are 10 numbers.
3. Mean:
   $\text{Mean}=\frac{621}{10}=62.1$$\text{Mean}=\frac{621}{10}=62.1$“Mean”=(621)/(10)=62.1\text{Mean} = \frac{621}{10} = 62.1$\text{Mean}=\frac{621}{10}=62.1$

2. Median:

1. Arrange the numbers in ascending order:
   45, 48, 50, 50, 58, 65, 68, 72, 75, 90
2. Since there are 10 numbers (even count), the median is the average of the 5th and 6th numbers:
   • The 5th number is 58.
   • The 6th number is 65.
3. Median:
   $\text{Median}=\frac{58+65}{2}=\frac{123}{2}=61.5$$\text{Median}=\frac{58+65}{2}=\frac{123}{2}=61.5$“Median”=(58+65)/(2)=(123)/(2)=61.5\text{Median} = \frac{58 + 65}{2} = \frac{123}{2} = 61.5$\text{Median}=\frac{58+65}{2}=\frac{123}{2}=61.5$

3. Mode:

1. In the given data set:
   50, 68, 90, 45, 75, 48, 72, 65, 58, 50
2. The number 50 appears twice, which is more frequent than any other number.

Summary:

• Mean: 62.1
• Median: 61.5
• Mode: 50

Explain the merits, limitations and uses of standard deviation.

Answer:

Merits of Standard Deviation:

1. Precision in Measuring Spread:
   Standard deviation gives a precise measure of how much individual data points deviate from the mean. It considers all data points, unlike simpler measures such as range, which only uses the extremes.
2. Comparison Across Data Sets:
   It enables comparison between different data sets to determine which set has greater variability or dispersion. A lower standard deviation indicates that the data points are closer to the mean, while a higher value shows more spread.
3. Useful in Many Fields:
   Standard deviation is widely used in various fields such as finance, economics, psychology, and the natural sciences to measure the risk, variability, or uncertainty of data sets.
4. Foundation for Other Statistical Tools:
   Many other statistical measures, such as variance, z-scores, and hypothesis testing, are based on the standard deviation. It also plays a key role in inferential statistics.
5. Applicable to Normal Distribution:
   For normally distributed data, standard deviation helps in defining the probability of data points falling within certain intervals, making it essential for predictions and probability calculations.

Limitations of Standard Deviation:

1. Sensitivity to Outliers:
   Standard deviation can be heavily affected by outliers, as extreme values can significantly increase the calculated spread of the data.
2. Assumes Normal Distribution:
   Standard deviation is most meaningful when data follows a normal (bell-curve) distribution. In cases of skewed or irregularly distributed data, its interpretation becomes less effective.
3. Not Easy to Interpret for All Data Sets:
   For those not familiar with statistical analysis, the concept of standard deviation may be difficult to grasp. Moreover, in some real-world contexts, it may not provide intuitive or actionable insights.
4. Relies on Mean:
   Since standard deviation is based on the mean, it inherits the limitations of the mean, such as not being a good measure of central tendency in cases of highly skewed distributions.
5. Difficult to Calculate Manually:
   For large datasets, calculating the standard deviation manually can be tedious and error-prone. While tools exist to compute it, the formula itself can be cumbersome without computational assistance.

Uses of Standard Deviation:

1. Risk Assessment in Finance:
   In financial markets, standard deviation is commonly used to measure the volatility of stocks or investment portfolios. A higher standard deviation signifies greater risk due to higher variability in returns.
2. Quality Control in Manufacturing:
   In production processes, standard deviation helps monitor product consistency. If the standard deviation of a product’s features is small, it implies uniformity, whereas a large deviation may indicate defects.
3. Education and Psychology:
   Standard deviation is used to understand the variability in test scores, intelligence quotients (IQ), or other psychological assessments. It shows how much individual performance deviates from the average.
4. Weather Forecasting:
   Meteorologists use standard deviation to analyze temperature fluctuations, helping them predict extreme weather conditions and establish climate patterns.
5. Scientific Research:
   In experiments, standard deviation is used to quantify variability in data and to understand the precision and reliability of measurements or observations.

Compute average deviation for the following data:

Answer:

Step 1: List the given data

Step 2: Calculate the mean (average)

Step 3: Calculate the deviation of each value from the mean

Step 4: Find the sum of all absolute deviations

Step 5: Calculate the average deviation

Conclusion:

Describe the properties of Normal Distribution Curve (NPC).

Answer:

1. Symmetry: The NPC is symmetric around its mean. This means the left and right sides of the curve are mirror images of each other. The mean, median, and mode of the distribution are all located at the same point, right at the center of the curve.
2. Bell-shaped: The curve has a characteristic bell shape. It starts low, rises to a peak at the mean, and then gradually tapers off symmetrically as it moves away from the center.
3. Mean, Median, Mode: In a perfectly normal distribution, the mean, median, and mode are equal and located at the center of the distribution. These measures of central tendency coincide at the peak.
4. Asymptotic: The tails of the normal distribution curve approach but never touch the horizontal axis. This property implies that, theoretically, the curve extends infinitely in both directions.
5. Empirical Rule (68-95-99.7 rule):
   • 68% of the data falls within one standard deviation of the mean (i.e., between μ – σ and μ + σ).
   • 95% of the data falls within two standard deviations of the mean (i.e., between μ – 2σ and μ + 2σ).
   • 99.7% of the data falls within three standard deviations of the mean (i.e., between μ – 3σ and μ + 3σ).
6. Unimodal: The normal distribution curve has only one peak (mode), making it unimodal.
7. Standard Normal Distribution: A special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. This allows for comparison of different normal distributions by converting data points into z-scores (standardized scores).
8. Probability Density: The area under the NPC represents probability, and the total area under the curve sums to 1 (or 100%). The height of the curve at any given point represents the likelihood of the random variable taking on that particular value.

Discuss the concept of Kurtosis and Skewness.

Answer:

1. Skewness

• Positive Skewness (Right-Skewed Distribution): When the right tail (the larger values) of the distribution is longer or fatter than the left tail. In this case, most of the data is concentrated on the left, and the mean is typically greater than the median.
• Negative Skewness (Left-Skewed Distribution): When the left tail (the smaller values) is longer or fatter than the right tail. Here, most of the data is concentrated on the right, and the mean is usually less than the median.
• Zero Skewness (Symmetrical Distribution): In a perfectly symmetrical distribution, the mean, median, and mode are equal, and there is no skewness. For example, the normal distribution has zero skewness.

Formula for Skewness:

• $X$$X$XX$X$ is a random variable
• $\mu$$\mu$mu\mu$\mu$ is the mean of the distribution
• $\sigma$$\sigma$sigma\sigma$\sigma$ is the standard deviation
• $E$$E$EE$E$ represents the expected value

2. Kurtosis

• High Kurtosis (Leptokurtic): Distributions with high kurtosis have heavy tails and a sharp peak, indicating more frequent extreme outliers. The data is highly concentrated around the mean, with more extreme values in the tails.
• Low Kurtosis (Platykurtic): Distributions with low kurtosis have lighter tails and a flatter peak. This indicates that the data points are more spread out and there are fewer extreme values.
• Normal Kurtosis (Mesokurtic): A distribution with kurtosis similar to a normal distribution has a kurtosis value of 3 (this is often subtracted by 3 to give a "zero kurtosis" interpretation). In this case, the distribution has moderate tails and no significant outliers.

Formula for Kurtosis:

• $X$$X$XX$X$, $\mu$$\mu$mu\mu$\mu$, and $\sigma$$\sigma$sigma\sigma$\sigma$ have the same meanings as in the skewness formula.

Interpretation of Kurtosis:

• Excess Kurtosis: To make kurtosis interpretation easier, we often subtract 3 from the kurtosis value (since a normal distribution has a kurtosis of 3). This is called excess kurtosis. Therefore:
  • Positive excess kurtosis: Indicates a leptokurtic distribution (heavy tails).
  • Negative excess kurtosis: Indicates a platykurtic distribution (light tails).

Summary:

• Skewness tells us about the asymmetry of the distribution (left or right-skewed).
• Kurtosis tells us about the distribution’s tails (whether they are heavy or light compared to a normal distribution).