Stats Final

If a small segment of the population is sampled then an estimate will be less precise.

TRUE

The standard deviation of X⎯⎯⎯X¯ suggests that the variation between observations is smaller than the variation between averages.

FALSE The standard deviation of the sample mean (standard error of the sample mean) suggests the variation between averages is less than the variation between observations.

The t distribution table lists tdf values for selected lower-tail probabilities and degrees of freedom df.

FALSE The t distribution table lists tdf values for selected upper-tail probabilities and degrees of freedom df.

When a statistic is used to estimate a parameter, the statistic is referred to as an estimator. A particular value of the estimator is called an estimate.

TRUE The statistic is used to estimate a parameter.

The labor force participation rate is the number of people in the labor force divided by the number of people in the country who are of working age and not institutionalized. The BLS reported in February 2012 that the labor force participation rate in the United States was 65.3% (Calculatedrisk.com). A marketing company asks 150 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force. What are the expected value and the standard error for a labor participation rate in the company's sample?

0.653 and 0.0389 Correct 76.44 and 0.0019 76.44 and 0.0389 0.653 and 0.0019 Explanation The expected value of PP is computed as E(P⎯⎯⎯)=pEP¯=p . The standard error of PP is computed as se(P⎯⎯⎯)=p(1−p)n‾‾‾‾‾‾√seP¯=p1-pn . E(p⎯⎯)Ep¯ = 0.653 SE(p⎯⎯)SEp¯ =SQRT(0.653*(1-0.653)/150) = 0.0389.

A Type I error is committed when we reject the null hypothesis, which is actually true.

TRUE A Type I error is committed when we reject the null hypothesis, which is actually true.

A simple random sample is a sample of n observations that has the same probability of being selected from the population as any other sample of n observations.

TRUE Most statistical methods presume simple random sample.

The null hypothesis typically corresponds to a presumed default state of nature.

TRUE We think of the null hypothesis as corresponding to a presumed default state of nature or status quo.

If we want to find the required sample size for the interval estimation of the population proportion, and no reasonable estimate of this proportion is available, we assume the worst-case scenario under which p^=0.5.p^⁢ = 0.5⁢.

TRUE If no other reasonable estimate of the population proportion is available, p^=0.5p^⁢ = 0.5 can be used as a conservative estimate to derive the optimal sample size

The required sample size for the interval estimation of the population mean can be computed if we specify the population standard deviation σ, the value of zα/2zα/2 based on the confidence level 100(1 − α)% and the desired margin of error E.

TRUE When estimating a 100(1 − α)% confidence interval for the population mean for a desired margin of error E, the minimum sample size n is computed by the formula: n = (zα/2σˆE)2zα/2σ^E2

The tdf distribution consists of a family of distributions where the actual shape of each one depends on the degrees of freedom, df. For lower values of df, the tdf distribution is similar to the zdistribution.

FALSE The tdf distribution consists of a family of distributions where the actual shape of each one depends on the degrees of freedom, df. As df increases, the tdf distribution becomes more similar to the z distribution; it is identical to the z distribution when df is infinity.

As a general guideline, we use the alternative hypothesis as a vehicle to establish something new, or contest the status quo, for which a corrective action may be required.

TRUE As a general guideline, we use the alternative hypothesis as a vehicle to establish something new, or contest the status quo, for which a corrective action may be required.

For a given sample size, any attempt to reduce the likelihood of making one type of error (Type I or Type II) will increase the likelihood of the other error.

TRUE It is not always easy to determine which of two errors has more serious consequences. For a given evidence, there is a trade-off between these errors; by reducing Type I error, we implicitly increase Type II error, and vice versa.

Bias refers to the tendency of a sample statistic to systematically over - or underestimate a population parameter.

TRUE When the information from a sample is not typical of information in the population in a systematic way, we say that bias has occurred.

A random sample of size 49 is taken from a population with mean µ= 26 and standard deviation σ = 7. The probability that the sample mean is greater than 27 is ______.

0.5675 Incorrect 0.8413 0.4325 0.1587 Correct Explanation If XX is normal, we can transform it into a standard normal random variable as Z=X−μσ/n√,Z=X-μσ/n, and any value of xx on XX has a corresponding value z on Z given by Z=x−μσ/n√.Z=x-μσ/n. Compute P(X>27).PX>27. Note that P(Z > z) = 1 − P(Z ≤ z). Use z table. The appropriate Excel function is =1-NORM.DIST(27,26,7/SQRT(49),TRUE) = 0.1587.

The average rent in a city is $1,510 per month with a standard deviation of $210. Assume rent follows the normal distribution. [You may find it useful to reference the z table.]a. What percentage of rents are between $1,300 and $1,720? (Round your answer to 2 decimal places.)b. What percentage of rents are less than $1,300? (Round your answer to 2 decimal places.)c. What percentage of rents are greater than $1,930? (Round your answer to 2 decimal places.) Explanation Let X represent rent in a city. a. P(1,300<X<1,720)=P(1,300−1,510210<Z<1,720−1,510210)P1,300<X<1,720=P1,300-1,510210<Z<1,720-1,510210 = P(−1 < Z < 1) = 0.6826, or 68.26% b. P(X<1,300)=P(Z<1,300−1,510210)PX<1,300=PZ<1,300-1,510210 = P(Z < −1) = 0.1587, or 15.87%c. P(X>1,930)=P(Z<1,930−1,510210)PX>1,930=PZ<1,930-1,510210 = P(Z > 2) = 0.0228,or 2.28%

EXCEL

Like the z distribution, the tdf distribution is symmetric around 0, bell-shaped, and with tails that approach the horizontal axis and eventually cross it.

FALSE Like the z distribution, the tdf distribution is symmetric around 0, bell-shaped, and with asymptotic tails—that is, the tails get closer and closer to the horizontal axis but never touch it.

The sampling distribution of P⎯⎯⎯P¯ is based on a normal distribution on samples of any size.

FALSE The sampling distribution of P⎯⎯⎯P¯ is based on a binomial distribution and can be approximated by a normal distribution for large samples.

In stratified random sampling, the population is first divided up into mutually exclusive and collectively exhaustive groups, called strata. A stratified sample includes randomly selected observations from each stratum, which are proportional to the stratum's size.

TRUE With stratified random sampling, the population is divided into groups based on one or more classification criteria. Then simple random samples are drawn from each group in sizes proportional to the relative size of each group in the population.