Location problems in statistics typically focus on estimating:

• A) The shape of the distribution.
• B) The probability distribution of the sample.
• C) The center or central tendency of the distribution.
• D) The scale of the distribution.

The p-value in hypothesis testing is defined as:

• A) The probability of obtaining the observed sample mean.
• B) The probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.
• C) The probability that the null hypothesis is true.
• D) The probability of making a Type I error.

A small p-value (e.g., less than 0.05) suggests that

• A) The null hypothesis is true.
• B) The sample size is large.
• C) There is strong evidence against the null hypothesis.
• D) The test statistic is small.

In the case of a random sample from a normal distribution, the sample mean is considered a minimal sufficient statistic because:

• A) It is the simplest form of an estimator.
• B) It is a location statistic.
• C) It is unbiased.
• D) It contains all information needed to estimate the population mean and variance.

Consider a random sample from a normal distribution with known variance. What could be an ancillary statistic in this scenario?

• A) Sample size
• B) Sample mean
• C) Sample variance
• D) Sample median

In scale problems, which statistic is commonly used to estimate the scale parameter?

• A) Sample standard deviation
• B) Sample mean
• C) Sample median
• D) Sample variance

The critical region in hypothesis testing is determined by:

• A) The confidence level chosen for the test.
• B) The significance level (α).
• C) The sample mean and standard deviation.
• D) The observed value of the test statistic.

To find the minimal sufficient statistic, first we have to check that the given statistic is __________or not.

• A) Unbiasedness
• B) sufficient
• C) Consistent
• D) Efficient

In hypothesis testing, an ancillary statistic is useful because:

• A) It can help in finding the best critical region.
• B) It is used to compute the p-value.
• C) It helps to estimate the parameter.
• D) It has no role in the decision-making process.

In a two-tailed test at α = 0.05, the critical region is:

• A) The two tails with a combined probability of 0.05.
• B) The region between the two means.
• C) The two tails with a combined probability of 0.10.
• D) The central 95% of the distribution.

The main objective of solving a location problem is to estimate:

• A) The sample mean.
• B) The population standard deviation.
• C) The variance.
• D) The median of the distribution.

An ancillary statistic is defined as:

• A) A statistic that is independent of the parameter being estimated.
• B) A statistic that contains all the information about the parameter.
• C) A statistic that is sufficient for the parameter.
• D) A statistic that is always unbiased.

In other words, the estimator which minimizes the posterior expected loss for each X is called:

• A) Bayes estimator
• B) Bayes decision
• C) Both the Bayes estimator and Bayes decision
• D) Neither the Bayes estimator nor the Bayes decision

For a sample from a Poisson distribution, which statistic is minimal sufficient?

• A) Sample range.
• B) Sample variance.
• C) Sample median
• D) Sample mean.

Scale problems in statistics typically involve the estimation of:

• A) The location parameter.
• B) The population mean.
• C) The dispersion or spread of the data.
• D) The probability of a certain outcome.

In a one-sided hypothesis test with α = 0.05, the critical region is:

• A) The right 5% tail of the distribution.
• B) The left 5% tail of the distribution.
• C) The center of the distribution.
• D) The top 95% of the distribution

In a scenario where the population variance is known, which of the following is an example of an ancillary statistic?

• A) Sample size
• B) Sample mean.
• C) Sample median.
• D) Sample range.

In hypothesis testing, the best critical region is the set of outcomes:

• A) That maximizes the probability of rejecting the null hypothesis when it is false.
• B) That lead to the acceptance of the null hypothesis.
• C) That correspond to values where the test statistic is close to zero.
• D) That ensures the probability of a Type II error is minimized.

For a normal distribution with known variance, a UMP test for the mean is based on:

• A) The sample mean and sample variance.
• B) The sample median.
• C) The likelihood ratio.
• D) The confidence interval.

The UMP test for the variance of a normal distribution is based on:

• A) Sample standard deviation.
• B) Likelihood ratio.
• C) Sample mean.
• D) Sample variance.