Class 12 Maths Chapter 13 Probability – Exercise 13.12 NCERT Solutions

Introduction

Exercise 13.12 focuses on sampling distributions and applications of probability in statistics. Students learn how to calculate probabilities related to sample means, apply the Central Limit Theorem, and use normal approximation for binomial distributions. This exercise is essential for understanding inferential statistics and hypothesis testing.

Key Concepts

1. Sampling Distribution of Mean: If population mean = $\mu$, variance = ${\sigma }^2$, sample size = $n$, then

$${\mu }_{\bar{X}}=\mu ,{\sigma }_{\bar{X}}^2=\frac{{\sigma }^2}{n}$$

2. Central Limit Theorem (CLT): For large $n$, distribution of sample mean approaches normal distribution.

3. Normal Approximation to Binomial: If $X\sim B(n,p)$, then for large $n$,

$$X\approx N(np,npq)$$

4. Continuity Correction: Adjustment of $\pm 0.5$ when approximating discrete distributions by continuous.

Students Frequently Make Mistakes

• Forgetting to divide variance by $n$ for sample mean.

• Misinterpreting CLT conditions.

• Errors in continuity correction.

• Confusing population mean with sample mean.

• Skipping conversion to Z‑score.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. If population mean = 50, variance = 25, sample size = 25, find mean and variance of sample mean.

$${\mu }_{\bar{X}}=50,{\sigma }_{\bar{X}}^2=\frac{25}{25}=1$$

Q2. If population mean = 100, variance = 36, sample size = 9, find mean and standard deviation of sample mean.

$${\mu }_{\bar{X}}=100,{\sigma }_{\bar{X}}=\sqrt{\frac{36}{9}}=2$$

Q3. If population mean = 60, variance = 49, sample size = 49, find mean and variance of sample mean.

$${\mu }_{\bar{X}}=60,{\sigma }_{\bar{X}}^2=\frac{49}{49}=1$$

Q4. If $X\sim B(100,0.5)$, approximate mean and variance.

$$\mu =50,{\sigma }^2=25$$

Q5. If $X\sim B(200,0.4)$, approximate mean and variance.

$$\mu =80,{\sigma }^2=48$$

Q6. If $X\sim B(150,0.3)$, approximate mean and variance.

$$\mu =45,{\sigma }^2=31.5$$

Q7. If $X\sim B(80,0.25)$, approximate mean and variance.

$$\mu =20,{\sigma }^2=15$$

Q8. If $X\sim B(120,0.6)$, approximate mean and variance.

$$\mu =72,{\sigma }^2=28.8$$

Q9. If $X\sim B(60,0.4)$, approximate mean and variance.

$$\mu =24,{\sigma }^2=14.4$$

Q10. If $X\sim B(100,0.3)$, approximate mean and variance.

$$\mu =30,{\sigma }^2=21$$

FAQs (10)

FAQ1. What is sampling distribution of mean? Distribution of sample mean values.

FAQ2. What is mean of sample mean distribution? Equal to population mean.

FAQ3. What is variance of sample mean distribution? ${\sigma }^2\mathrm{/}n$.

FAQ4. What is Central Limit Theorem? Sample mean tends to normal distribution for large $n$.

FAQ5. What is normal approximation to binomial? $X\sim B(n,p)$ approximated by $N(np,npq)$.

FAQ6. What is continuity correction? Adjustment of $\pm 0.5$ for discrete to continuous approximation.

FAQ7. What is mean of $B(100,0.5)$?

FAQ8. What is variance of $B(100,0.5)$?

FAQ9. What is mean of $B(200,0.4)$?

FAQ10. Why is Exercise 13.12 important? It builds mastery of sampling distributions and approximation techniques.

Conclusion

Exercise 13.12 has 10 solved questions and 10 FAQs that strengthen your understanding of sampling distributions and normal approximation to binomial distribution. This completes the Probability chapter in Class 12 Maths.

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