Problem Set 2 - Estimation and Hypothesis Testing#
(Check Canvas for due dates and office hours)
Answer case-by-case (use common notations).
If the population follows \(\mathcal{B}(N,p)\).
What is the distribution of \(\bar{x}\)?
What is \(\hat{p}\)?
If a sample of size 50 from a binomial distribution of 5 trials is known to have \(\bar{x} = 3.6\),
What is \(\hat{p}\)?
Provide 90% lower confidence bound on \(\hat{p}\).
If the population follows \(Poisson(\lambda)\).
What is the distribution of \(\bar{x}\)?
What is \(\hat{\lambda}\)?
Assuming the poulation follows a normal distribution \(\mathcal{N}(\mu,\sigma^2)\),
What are the distributions of
\(\bar{x}\)
\(\frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\)
\(\frac{\bar{x} - \mu}{s/\sqrt{n}}\)
\(\frac{(n-1)s^2}{\sigma^2}\)
What are estimates of
\(\hat{\mu}\)
\(\hat{\sigma^2}\)
\(\hat{\sigma}\)
If we choose a sample of size 9 from a normal distribution, provide confidence bounds on \(\hat{\mu}\) for each case.
95% two-sided bounds with \(\bar{x} = 1.5\) and \(\sigma^2 = 4\).
95% two-sided bounds with \(\bar{x} = 1.5\) and \(s^2 = 4\).
95% upper confidence bound with \(\bar{x} = 1.5\) and \(\sigma^2 = 4\).
95% lower confidence bound with \(\bar{x} = 1.5\) and \(s^2 = 4\).
If we choose a sample of size 9 from a normal distribution with \(\mu = 1.5\) and \(\sigma^2 = 4\).
What is the probability (as percentage) that \(\bar{x} \geq 2.327\)?
What is the probability (as percentage) that \(\bar{x} \leq 0.26\)?
What is the probability (as percentage) that \( -0.737 \leq \bar{x} \leq 3.737\)?
If we choose a sample of size 9 with \(s^2 = 4\) from a normal distribution with \(\mu = 1.5\) and unknown \(\sigma^2\).
What is the probability (as percentage) that \(\bar{x} \geq 2.327\)?
What is the probability (as percentage) that \(\bar{x} \leq 0.26\)?
What is the probability (as percentage) that \( -0.737 \leq \bar{x} \leq 3.737\)?
Comment on the difference between answers of this question and the previous.
Answer Qualitatively: If we choose a sample of size 9 with \(\bar{x} = 1.5\), arrange the three statements in order from more likely to less likely. (NO NEED TO CALCULATE ANY VALUES).
(1) Sample belongs to a population with \(\mu = 2\) and \(\sigma^2 = 0.01\)?
(2) Sample belongs to a population with \(\mu = 2\) and \(\sigma^2 = 1\)?
(3) Sample belongs to a population with \(\mu = 0\) and \(\sigma^2 = 1\)?
Forming Hypothesis (just write \(H_0\) and \(H_a\), no need to analyze): You have a sample of 9 softdrink cans with mean width \(\bar{x} = 1.5\) (inches) from a normal distributed manufacturing process (with huge number of cans) with unknown \(\mu\). Convert the following plain language statements into null (\(H_0\)) and alternative hypotheses (\(H_a\)).
Can the manufacturing process have a mean width that is different from \(\hat{\mu} \,\, (=\bar{x})\)?
Can the manufacturing process have a mean width that is greater than \(3.737\)?
Can the manufacturing process have a mean width that is less than \(0.26\)?
If we choose a sample of size 9 with \(\bar{x} = 1.5\) from a normal distribution with unknown \(\mu\). Consider Type-I error rate (\(\alpha\)) as 5% or 10%.
What is the p-value for \(H_0: \mu = 2\); \(H_a: \mu \neq 2\) with known \(\sigma^2 = 0.01\)? What will be the result of hypothesis testing?
What is the p-value for \(H_0: \mu = 2\); \(H_a: \mu \neq 2\) with \(s^2 = 0.01\) and unknown \(\sigma^2\)? What will be the result of hypothesis testing?
What is the p-value for \(H_0: \mu = 2\); \(H_a: \mu < 2\) with known \(\sigma^2 = 1\)? What will be the result of hypothesis testing?
What is the p-value for \(H_0: \mu = 2\); \(H_a: \mu < 2\) with \(s^2 = 1\) and unknown \(\sigma^2\)? What will be the result of hypothesis testing?
What is the p-value for \(H_0: \mu = 0\); \(H_a: \mu < 0\) with known \(\sigma^2 = 1\)? What will be the result of hypothesis testing?
What is the p-value for \(H_0: \mu = 0\); \(H_a: \mu < 0\) with \(s^2 = 1\) and unknown \(\sigma^2\)? What will be the result of hypothesis testing?
Textbook Problem 4.3: The service life of a battery used in a cardiac pacemaker is assumed to be normally distributed. A random sample of 10 batteries is subjected to an accelerated life test by running them continuously at an elevated temperature until failure, and the following lifetimes (in hours) are obtained: 25.5, 26.1, 26.8, 23.2, 24.2, 28.4, 25.0, 27.8, 27.3, and 25.7.
The manufacturer wants to be certain that the mean battery life exceeds 25 hours. What conclusions can be drawn from these data (use \(\alpha\) = 0.05)?
Construct a 90% two-sided confidence interval on mean life in the accelerated test.
Textbook Problem 4.14: A random sample of 200 printed circuit boards contains 18 defective or nonconforming units. Estimate the process fraction nonconforming.
Test the hypothesis that the true fraction nonconforming in this process is 0.10. Use \(\alpha= 0.05\). Find the P-value.
Construct a 90% two-sided confidence interval on the true fraction nonconforming in the production process.
Textbook Problem 4.9: The output voltage of a power supply is assumed to be normally distributed. Sixteen observations taken at random on voltage are as follows: 10.35, 9.30, 10.00, 9.96, 11.65, 12.00, 11.25, 9.58, 11.54, 9.95, 10.28, 8.37, 10.44, 9.25, 9.38, and 10.85.
Test the hypothesis that \(\sigma^2 = 1.0\) using \(\alpha= 0.05\).
Construct a 95% two-sided confidence interval on \(\sigma^2\).
Construct a 95% upper confidence interval on \(\sigma\).