Normal Distribution and Sampling

Introduction

As-salamu alaykum, students. Ustad Bilal here. In your A Level journey, few topics are as powerful and widely applicable as the Normal Distribution. You see it everywhere, from the heights of students in your college to the scores in a national exam like the NUST entry test. It's the famous "bell curve," and understanding it is not just about passing Paper 5 and 6; it's about understanding the very language of data and uncertainty.

In this chapter, we will first become experts in the Normal distribution itself – how to calculate probabilities and work backwards to find key parameters. Then, we will bridge the gap from theory to reality with Sampling. It’s impossible to survey every single person in Pakistan, but by taking a clever sample, we can make remarkably accurate statements about the entire population. We'll explore the Central Limit Theorem, a truly magical idea, which allows us to do this.

Finally, we'll use this knowledge to perform hypothesis tests and construct confidence intervals, the tools statisticians use to make decisions and express certainty. This is where you move from being a student of mathematics to a practitioner. So, get your calculators and formula sheets ready. Let's begin, bismillah.

Core Theory

The Normal distribution is a continuous probability distribution described by two parameters: the mean, μ (mu), and the variance, σ² (sigma squared). We write this as X ~ N(μ, σ²).

Properties of the Normal Distribution:

1. Symmetry: The curve is perfectly symmetrical about the mean, μ.
2. Bell Shape: It has a characteristic bell shape. The peak of the bell is at the mean.
3. Mean, Median, Mode: For a normal distribution, the mean, median, and mode are all equal.
4. Area: The total area under the curve is exactly 1, representing a total probability of 100%.
5. Asymptotic: The curve gets closer and closer to the horizontal axis but never touches it.

The Standard Normal Distribution (Z)

Calculating probabilities for any N(μ, σ²) would require infinite tables. To solve this, we 'standardise' any normal variable X to a special case called the Standard Normal Distribution, denoted by Z, which has a mean of 0 and a variance of 1.

Z ~ N(0, 1)

The formula for standardisation is fundamental:

Z = (X - μ) / σ

This formula converts your specific X-value into a Z-score, which tells you how many standard deviations your value is away from the mean. Once you have a Z-score, you can use the standard normal tables (in your MF19/MF20 formula book) to find probabilities. The tables give you P(Z < z), the cumulative probability up to a value z.

* For P(Z > a), you calculate 1 - P(Z < a).

* For P(a < Z < b), you calculate P(Z < b) - P(Z < a).

Example: If X ~ N(50, 16), find P(X < 55).

Here, μ = 50 and σ² = 16, so σ = 4.

Z = (55 - 50) / 4 = 1.25

P(X < 55) = P(Z < 1.25). From the tables, this is 0.8944.

Normal Approximation to Binomial

When a binomial distribution B(n, p) has a large 'n' and 'p' is not too close to 0 or 1 (a good rule is np > 5 and nq > 5, where q=1-p), its shape is very similar to a normal distribution.

We can approximate it using: Y ~ N(np, npq).

Crucially, since we are approximating a discrete distribution (Binomial) with a continuous one (Normal), we must use a continuity correction.

* P(X = k) ≈ P(k - 0.5 < Y < k + 0.5)

* P(X ≤ k) ≈ P(Y < k + 0.5)

* P(X < k) ≈ P(Y < k - 0.5)

* P(X ≥ k) ≈ P(Y > k - 0.5)

* P(X > k) ≈ P(Y > k + 0.5)

The Central Limit Theorem (CLT)

This is one of the most important theorems in all of statistics. It states that if you take a sufficiently large random sample (usually n > 30) from *any* population with mean μ and variance σ², the distribution of the sample means (X̄) will be approximately normal.

X̄ ~ N(μ, σ²/n)

Notice the variance is σ²/n, not σ². This makes sense: the average of a sample is less likely to be at the extremes than a single observation. The standard deviation of the sample mean, σ/√n, is called the standard error.

Confidence Intervals (CI) for the Population Mean μ

A CI gives a range of plausible values for an unknown population mean μ, based on a sample mean X̄. A 95% CI means we are 95% confident that the true population mean lies within this interval.

For a large sample, the formula is:

CI = X̄ ± z* (σ / √n)

Where:

* X̄ is the sample mean.

* σ is the population standard deviation (if unknown, we use the sample standard deviation 's' for large n).

* n is the sample size.

* z* is the critical z-value for the desired confidence level (e.g., for 95%, z* = 1.96; for 99%, z* = 2.576).

Hypothesis Testing for the Mean μ

This is a formal procedure to test a claim about a population mean.

1. State Hypotheses:

* Null Hypothesis (H₀): The status quo, e.g., H₀: μ = k.

* Alternative Hypothesis (H₁): What we want to prove, e.g., H₁: μ > k (one-tailed), H₁: μ < k (one-tailed), or H₁: μ ≠ k (two-tailed).

1. Calculate Test Statistic:

* z-test (if σ is known or n is large): z = (X̄ - μ) / (σ / √n)

1. Determine Critical Region / p-value:

* Critical Value Method: Compare your test statistic to the critical value from the Z-table at your chosen significance level (α, e.g., 5% or 0.05). If it falls in the critical region, reject H₀.

* p-value Method: Calculate the p-value, which is the probability of getting a result as extreme as your sample. If p-value < α, reject H₀.

1. Conclusion: State your conclusion in the context of the problem, e.g., "There is sufficient evidence at the 5% significance level to suggest the mean score has increased."

Type I and Type II Errors

* Type I Error: Rejecting H₀ when H₀ is actually true. The probability of this is P(Type I Error) = α.

* Type II Error: Failing to reject H₀ when H₀ is actually false. The probability of this is denoted by β.

Key Definitions

* Normal Distribution N(μ, σ²): A continuous probability distribution defined by its mean (μ) and variance (σ²).

* Standardisation: The process of converting an X ~ N(μ, σ²) variable to a Z ~ N(0, 1) variable using the formula Z = (X - μ) / σ.

* Continuity Correction: An adjustment made when approximating a discrete distribution (like Binomial) with a continuous one (Normal), e.g., P(X=k) becomes P(k-0.5 < Y < k+0.5).

* Central Limit Theorem (CLT): A theorem stating that the distribution of sample means (X̄) from any population is approximately normal for large sample sizes (n>30), with mean μ and variance σ²/n.

* Standard Error: The standard deviation of a sampling distribution, typically of the sample mean. It is calculated as σ/√n.

* Confidence Interval: An interval estimate for a population parameter. For the mean, it's typically X̄ ± (critical value) × (standard error).

* Hypothesis Test: A statistical procedure to test a claim (H₀) against an alternative (H₁).

* p-value: The probability, assuming H₀ is true, of observing a test statistic at least as extreme as the one calculated from the sample data.

* Significance Level (α): The probability of making a Type I error. It's the threshold for rejecting the null hypothesis (reject H₀ if p-value < α).

* Type I Error: Rejecting a true null hypothesis.

* Type II Error: Failing to reject a false null hypothesis.

Worked Examples (Pakistani Context)

Example 1: Finding μ and σ

The time, X minutes, taken by students to complete a statistics quiz is normally distributed. It is found that 10% of students take less than 15 minutes, and 5% of students take more than 30 minutes. Find the mean (μ) and standard deviation (σ) of the time taken.

Solution:

Let X ~ N(μ, σ²). We are given:

1. P(X < 15) = 0.10
2. P(X > 30) = 0.05 => P(X < 30) = 0.95

Standardise both equations:

1. P(Z < (15 - μ) / σ) = 0.10
2. P(Z < (30 - μ) / σ) = 0.95

Now, we use the inverse normal function on our calculators or look up these probabilities in the Z-table.

For P(Z < z₁) = 0.10, the area is in the left tail. The table gives areas > 0.5, so we look for P(Z < -z₁) = 0.90. This is not standard. Instead, we use the symmetry property. P(Z < z₁) = 0.10 implies z₁ is negative. We find the z-value corresponding to an upper tail of 0.10, which is the same as a cumulative probability of 0.90. From tables, z corresponding to 0.90 is approximately 1.282. So, z₁ = -1.282.

(15 - μ) / σ = -1.282 => 15 - μ = -1.282σ (Equation A)

For P(Z < z₂) = 0.95, from the tables, z₂ = 1.645.

(30 - μ) / σ = 1.645 => 30 - μ = 1.645σ (Equation B)

Now we have a system of two simultaneous linear equations:

(B) - (A):

(30 - μ) - (15 - μ) = 1.645σ - (-1.282σ)

15 = 2.927σ

σ = 15 / 2.927 = 5.125 (3 d.p.)

Substitute σ back into Equation B:

30 - μ = 1.645 * (5.125)

30 - μ = 8.431

μ = 30 - 8.431 = 21.569 (3 d.p.)

So, the mean time is 21.6 minutes and the standard deviation is 5.13 minutes (to 3 s.f.).

Example 2: Hypothesis Test on Lahore Metro Commute Times

The Lahore Metro Orange Line authority claims the average commute time between two major stations is 25 minutes. A transport analyst believes the time has increased due to recent signal upgrades. They take a random sample of 50 commute times and find a sample mean of 26.5 minutes. Assuming the population standard deviation of commute times is known to be 4 minutes, test the analyst's belief at the 5% significance level.

Solution:

Step 1: State Hypotheses

Let μ be the true mean commute time.

Null Hypothesis H₀: μ = 25 (The mean time is still 25 minutes)

Alternative Hypothesis H₁: μ > 25 (The mean time has increased)

This is a one-tailed test.

Step 2: Significance Level

α = 0.05

Step 3: Calculate the Test Statistic

We have: n = 50, X̄ = 26.5, σ = 4, μ₀ = 25.

Since n > 30, we can use the z-test.

The standard error (SE) = σ / √n = 4 / √50 = 0.5657

z = (X̄ - μ) / SE = (26.5 - 25) / 0.5657 = 1.5 / 0.5657 = 2.652

Step 4: Determine Critical Region / p-value

* Critical Value Method: For a one-tailed test at α = 0.05, the critical value of z is found by looking for the z-score with 0.95 area to its left. z_crit = 1.645.

Our test statistic z = 2.652 is greater than 1.645, so it falls in the critical region.

* p-value Method: The p-value is P(Z > 2.652).

p-value = 1 - P(Z < 2.652) = 1 - 0.9960 = 0.0040.

Step 5: Make a Decision

* Using the critical value method: Since 2.652 > 1.645, we reject H₀.

* Using the p-value method: Since p-value (0.0040) < α (0.05), we reject H₀.

Step 6: Conclusion

Both methods lead to the same decision. There is sufficient evidence at the 5% significance level to reject the null hypothesis. We conclude that the mean commute time on the Lahore Metro Orange Line has increased.

Exam Technique

Beta, how you present your answer is as important as getting it right. Cambridge examiners look for clear, logical steps.

* Show Your Work: For any probability calculation, always write down the standardisation step: P(X < a) = P(Z < (a - μ)/σ). This single line can get you a method mark even if your final answer is wrong.

* Diagrams are Your Friend: For tricky probability questions (like P(a < X < b) or finding μ/σ), sketch a quick bell curve. Shade the area you need. It helps you visualise the problem and avoid simple mistakes like subtracting probabilities the wrong way.

* Continuity Correction: This is the most common mistake in Normal approximations to Binomial. Write it down clearly. For P(X ≥ 10), write "Applying continuity correction, we need P(Y > 9.5)" where Y is the normal variable. Examiners need to see you've consciously applied it.

* Context is King: In hypothesis testing and confidence interval questions, always write your final conclusion back in the context of the original problem. Don't just say "Reject H₀." Say, "There is sufficient evidence to suggest the mean weight of the mangoes has decreased."

* Read the Question Carefully: Distinguish between population variance (σ²) and sample variance (s²), and between the distribution of X and the distribution of the sample mean X̄. The variance of X̄ is σ²/n. This is a classic trip-up.

* Precision: Use at least 4 decimal places for probabilities from the table and for your intermediate z-values. Round your final answer to 3 significant figures as specified in the question paper instructions unless told otherwise.

* Hypothesis Testing Structure: Always lay out your hypothesis tests with the 5 or 6 steps: H₀/H₁, significance level, test statistic calculation, critical value/p-value, decision, and contextual conclusion. This structure ensures you don't miss any marks. Insha'Allah, following these tips will secure you top marks.