Inferential Statistics

Interval Estimation

Q1. How does interval estimation overcome limitation of point estimation?

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Interval Estimation

Q2. Suppose a random sample of size n is taken from a normal population of values for a quantitative variable whose mean ($μ$) is unknown, when the standard deviation ($σ$) is given. A $95%$ confidence interval (CI) for $μ$ is:

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Interval Estimation

Q3. How should we interpret the $95%$ CI for a population mean($\mu$)?

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Interval Estimation

Q4. The IQ level of students at a particular university has an unknown mean, $μ$, and a known standard deviation, $σ = 15$. A simple random sample of $100$ students is found to have a sample mean IQ, $\hat{x} = 115$. Estimate $μ$ with $90%$, $95%$, and $99%$ confidence intervals.

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Interval Estimation

Q5. Explain the trade-off between the level of the confidence and the precision with which the parameter is estimated?

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Interval Estimation

Q6. Write the general structure of the confidence intervals.

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Interval Estimation

Q7. Explain margin of error in interval estimation. What value does it encode?

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Interval Estimation

Q8. How can we reduce margin or error $m$ without compromising on the level of confidence?

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Interval Estimation

Q9. Find the general expression for the required $n$ for a desired margin of error $m$ and certain level of confidence.

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Interval Estimation

Q10. Suppose that based on a random sample, a $95%$ confidence interval for the mean hours slept (per day) among graduate students was found to be $(6.5, 6.9)$. What is the margin of error of this confidence interval?

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Interval Estimation

Q11. IQ scores are known to vary normally with a standard deviation of $15 $. How many students should be sampled if we want to estimate the population mean IQ at$99%$ confidence with a margin of error equal to $2$?

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Interval Estimation

Q12. In which case it is not safe to use confidence interval developed using CLT?

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Variable varies normaly and sample size is small( $n < 30$ )
Variable varies normaly and sample size is large
Variable does not vary normal and sample size is small
Variable does not vary normal and sample size is large

Interval Estimation

Q13. How should we calculate confidence interval when the population standard deviation $\sigma$ is not known?

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Interval Estimation

Q14. <b>True/False</b> For large values of $n $, the$ t^$multipliers are not much different from the$ z^$ multipliers?

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Interval Estimation

Q15. Write the expression of confidence interval when variable of interest is categorical?

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Interval Estimation

Q16. A poll asked a random sample of $1,000$ U.S. adults, "Do you think that the use of marijuana should be legalized?"$560$ of those asked answered yes.

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Based on the poll's results, estimate p, the proportion of all U.S. adults who believe the use of marijuana should be legalized, with a 95% confidence interval.
Give an interpretation of the margin of error in context.
Do the results of this poll give evidence that the majority of U.S. adults believe that the use of marijuana should be legalized?

Interval Estimation

Q17. Under what condition we can use to construct CI in case of estimating $p$ using $z^*$?

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Interval Estimation

Q18. Suppose that you take $100$ random newborn puppies and determine that the average weight is $1$ pound with the population standard deviation of $0.12$ pounds. Assuming the weight of newborn puppies follows a normal distribution, calculate the $95\\%$ confidence interval for the average weight of all newborn puppies.

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Interval Estimation

Q19. Suppose that we examine $100$ newborn puppies and the $95%$ confidence interval for their average weight is $[0.9, 1.1]$ pounds. Which of the following statements is true?

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Given a random newborn puppy, its weight has a $95%$ chance of being between $0.9$ and $1.1$ pounds.
If we examine another $100$ newborn puppies, their mean has a $95%$ chance of being in that interval.
We're $95\\%$ confident that this interval captured the true mean weight.

Interval Estimation

Q20. Suppose we have a random variable X supported on $[0,1]$ from which we can draw samples. How can we come up with an unbiased estimate of the median of X?

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Interval Estimation

Q21. The weight of newborn puppies is roughly symmetric with a mean of 1 pound and a standard deviation of 0.12. Your favorite newborn puppy weighs 1.1 pounds.

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Calculate your puppy’s z-score (standard score).
1. How much does your newborn puppy have to weigh to be in the top 10% in terms of weight?
2. Suppose the weight of newborn puppies followed a skewed distribution. Would it still make sense to calculate z-scores?

Interval Estimation

Q22. When should you use a Z-Test instead of a T-Test?

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Q1. How does interval estimation overcome limitation of point estimation?

Q2. Suppose a random sample of size n is taken from a normal population of values for a quantitative variable whose mean ($μ$) is unknown, when the standard deviation ($σ$) is given. A $95%$ confidence interval (CI) for $μ$ is:

Q3. How should we interpret the $95%$ CI for a population mean($\mu$)?

Q4. The IQ level of students at a particular university has an unknown mean, $μ$, and a known standard deviation, $σ = 15$. A simple random sample of $100$ students is found to have a sample mean IQ, $\hat{x} = 115$. Estimate $μ$ with $90%$, $95%$, and $99%$ confidence intervals.

Q5. Explain the trade-off between the level of the confidence and the precision with which the parameter is estimated?

Q6. Write the general structure of the confidence intervals.

Q7. Explain margin of error in interval estimation. What value does it encode?

Q8. How can we reduce margin or error $m$ without compromising on the level of confidence?

Q9. Find the general expression for the required $n$ for a desired margin of error $m$ and certain level of confidence.

Q10. Suppose that based on a random sample, a $95%$ confidence interval for the mean hours slept (per day) among graduate students was found to be $(6.5, 6.9)$. What is the margin of error of this confidence interval?

Q11. IQ scores are known to vary normally with a standard deviation of $15 $. How many students should be sampled if we want to estimate the population mean IQ at$99%$ confidence with a margin of error equal to $2$?

Q12. In which case it is not safe to use confidence interval developed using CLT?

Q13. How should we calculate confidence interval when the population standard deviation $\sigma$ is not known?

Q14. <b>True/False</b> For large values of $n $, the$ t^*$multipliers are not much different from the$ z^*$ multipliers?

Q15. Write the expression of confidence interval when variable of interest is categorical?

Q16. A poll asked a random sample of $1,000$ U.S. adults, "Do you think that the use of marijuana should be legalized?"$560$ of those asked answered yes.

Q17. Under what condition we can use to construct CI in case of estimating $p$ using $z^*$?

Q19. Suppose that we examine $100$ newborn puppies and the $95%$ confidence interval for their average weight is $[0.9, 1.1]$ pounds. Which of the following statements is true?

Q20. Suppose we have a random variable X supported on $[0,1]$ from which we can draw samples. How can we come up with an unbiased estimate of the median of X?

Q21. The weight of newborn puppies is roughly symmetric with a mean of 1 pound and a standard deviation of 0.12. Your favorite newborn puppy weighs 1.1 pounds.

Q22. When should you use a Z-Test instead of a T-Test?

Q14. <b>True/False</b> For large values of $n $, the$ t^$multipliers are not much different from the$ z^$ multipliers?