Mention the sample size that is required to claim that the distribution of sample means x ¯ to be approximately normal.

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3. Bayesian Inference – Updating Beliefs A medical test for a rare disease has the following characteristics: Sensitivity (true positive rate): 99% Specificity (true negative rate): 98% The disease occurs in 0.5% of the population. A patient receives a positive test result. Questions: a) Define the relevant events and use Bayes’ Theorem to compute the probability that the patient actually has the disease.b) Explain why the result might seem counterintuitive, despite the high sensitivity and specificity.c) Discuss how prior probabilities influence posterior beliefs in Bayesian inference.d) Suppose a second, independent test with the same accuracy is conducted and is also positive. Update the probability that the patient has the disease.

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

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3. Bayesian Inference – Updating Beliefs A medical test for a rare disease has the following characteristics: Sensitivity (true positive rate): 99% Specificity (true negative rate): 98% The disease occurs in 0.5% of the population. A patient receives a positive test result. Questions: a) Define the relevant events and use Bayes’ Theorem to compute the probability that the patient actually has the disease.b) Explain why the result might seem counterintuitive, despite the high sensitivity and specificity.c) Discuss how prior probabilities influence posterior beliefs in Bayesian inference.d) Suppose a second, independent test with the same accuracy is conducted and is also positive. Update the probability that the patient has the disease.

4. Linear Regression - Model Assumptions and Interpretation A real estate analyst is studying how house prices (Y) are related to house size in square feet (X). A simple linear regression model is proposed: The analyst fits the model and obtains: • Ŷ50,000+150X YBoB₁X + € • R² = 0.76 • Residuals show a fan-shaped pattern when plotted against fitted values. Questions: a) Interpret the slope coefficient in context. b) Explain what the R² value tells us about the model's performance. c) Based on the residual pattern, what regression assumption is likely violated? What might be the consequence? d) Suggest at least two remedies to improve the model, based on the residual analysis.

5. Probability Distributions – Continuous Random Variables A factory machine produces metal rods whose lengths (in cm) follow a continuous uniform distribution on the interval [98, 102]. Questions: a) Define the probability density function (PDF) of the rod length.b) Calculate the probability that a randomly selected rod is shorter than 99 cm.c) Determine the expected value and variance of rod lengths.d) If a sample of 25 rods is selected, what is the probability that their average length is between 99.5 cm and 100.5 cm? Justify your answer using the appropriate distribution.

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

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Answer 3

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

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Answer 5

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

Answer 6

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

Answer 7

Chapter 6.5, Problem 7P

(a)

To determine

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

Answer 8

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

Answer 9

(b)

To determine

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

Answer 10

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Mention the sample size that is required to claim that the distribution of sample means x ¯ to be approximately normal.

Solution Summary: The author explains the sample size that is required to claim that the distribution of sample is approximately normal, even if the population is not normally distributed.

Mention the sample size that is required to claim that the distribution of sample means x¯ to be approximately normal.

Identify whether it is necessary to make any restrictions about sample size for claiming that the distribution of sample means x¯ taken from random samples of a given size is normal, if the original distribution of x values is known to be normal.

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