Question

Question 1

Why can you never have 100% confidence in correctly estimating the population characteristic of interest?

When are you able to use the t distribution to develop the confidence interval estimation for the mean?

Why is it true that for a given sample size, n, an increase in confidence is acheived by widening (and making less precise) the confidence interval?

Why is the sample size needed to determine the proportion smaller when the population proportion is 0.20 than when the population proportion is 0.50?

What is the difference between a null hypothesis, H0, and an alternative hypothesis, H1?

What is the difference between a one-tailed test and a two-tailed test?

What is meant by a p value?

What is the six step critical value approach to hypthesis testing? List the six steps.

Answer 1

Why can you never have 100% confidence in correctly estimating the population characteristic of interest?

In statistics, a confidence interval (CI) is a kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient. More specifically, the meaning of the term "confidence level" is that, if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level; this is guaranteed by the reasoning underlying the construction of confidence intervals.[1][2][3]

Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter. However, in rare cases, none of these values may cover the value of the parameter. The level of confidence of the confidence interval would indicate the probability that the confidence range captures this true population parameter given a distribution of samples. It does not describe any single sample. This value is represented by a percentage, so when we say, "we are 99% confident that the true value of the parameter is in our confidence interval", we express that 99% of the observed confidence intervals will hold the true value of the parameter. After a sample is taken, the population parameter is either in the interval made or not, there is no chance. The level of confidence is set by the researcher (not determined by data) . If a corresponding hypothesis test is performed, the confidence level corresponds with the level of significance, i.e. a 95% confidence interval reflects an significance level of 0.05, and the confidence interval contains the parameter values that, when tested, should not be rejected with the same sample. Greater levels of confidence give larger confidence intervals, and hence less precise estimates of the parameter. Confidence intervals of difference parameters not containing 0 imply that that there is a statistically significant difference between the populations.

Certain factors may affect the confidence interval size including size of sample, level of confidence, and population variability. A larger sample size normally will lead to a better estimate of the population parameter.

A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained. (An interval intended to have such a property, called a credible interval, can be estimated using Bayesian methods; but such methods bring with them their own distinct strengths and weaknesses).

When are you able to use the t distribution to develop the confidence interval estimation for the mean?

Student’s t-distribution (or simply the t-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

Why is it true that for a given sample size, n, an increase in confidence is acheived by widening (and making less precise) the confidence interval?

(b) It is true that for a given sample size, n, increase in confidence is achieved by widening the confidence inter val. The reason is that when we collect a random sample a wider interval around sample mean has a higher prob- ability of containing the true mean. The value of this probability is same as confidence level for the confidence interval

Why is the sample size needed to determine the proportion smaller when the population proportion is 0.20 than when the population proportion is 0.50?

The sample size formula for p is: n = (z/ME)^2 * p * (1-p)

The required sample size is the largest when p = .5, because the quantity p * (1-p) is the largest when p = .5, i.e.

.1 * (1-.1) = .09
.3 * (1-.3) = .21
.5 * (1-.5) = .25
.7 * (1-.7) = .21
.9 * (1-.9) = .01