Business Statistics (True/False) II

The normal probability distribution is symmetric and bell-shaped.

True Explanation: The normal probability distribution is symmetric and bell-shaped.

In a one-tailed test, the rejection region is located under one tail (left or right) of the corresponding probability distribution, while in a two- tailed test this region is located under both tails.

True Explanation: A one-tailed test involves a null hypothesis that can be rejected only on one side of the hypothetical value. In a two-tailed test, we can reject the null hypothesis on either side of the hypothesized value of the population parameter.

A point estimator refers to an estimator that provides a single value.

True Explanation: A point estimator provides a single value or a point.

A Type 2 error is made when we reject the null hypothesis and the null hypothesis is actually false.

False Explanation: A Type II error is made when we do not reject the null hypothesis that is actually false.

Cumulative distribution functions can only be used to compute probabilities for continuous random variables

False Explanation: A cumulative distribution function F is computed as F(x) = P(X ≤ x).

Selection bias occurs when the sample is mistakenly divided into strata, and random samples are drawn from each stratum

False Explanation: Selection bias refers to a systematic exclusion of certain groups from consideration for the sample.

The alternative hypothesis typically agrees with the status quo.

False Explanation: The alternative hypothesis contradicts the default state of nature or status quo.

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/2 based on the confidence level 100(1 − α)% and the desired margin of error D.

True

The tdf distribution has broader tails than the z distribution

True Explanation: The tdf distribution has slightly broader tails than the z distribution.

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 Explanation: 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.

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-bar = 0.5.

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

A confidence interval provides a value that, with a certain measure of confidence, is the population parameter of interest

False Explanation: A confidence interval provides a range of values that, with a certain level of confidence, contains the population parameter of interest.

Cluster sampling is preferred when the objective is to increase precision.

False Explanation: Cluster sampling is preferred when the objective is to decrease costs. Stratified sample is preferred when the objective is to increase precision.

According the empirical rule for normally distributed variables, 75% of the values fall within one standard deviation of the mean

False Explanation: For normally distributed random variables the empirical rule states that 68.26% of the values fall within one standard deviation of the mean.

If the expected value of a sample mean equals the population mean, the sample mean is biased.

False Explanation: If the expected value of a sample mean equals the population mean, the sample mean is unbiased.

On the basis of sample information, we either "accept the null hypothesis" or "reject the null hypothesis."

False Explanation: On the basis of sample information, we either "reject the null hypothesis" or "do not reject the null hypothesis." Only one of two hypotheses is true and the hypotheses cover all possible values of the population parameter.

The central limit theorem approximation improves as the sample size decreases.

False Explanation: The central limit theorem approximation improves as the sample size increases.

The lognormal distribution is clearly negatively skewed for σ > 1.

False Explanation: The lognormal distribution is clearly positively skewed for σ > 1.

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

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

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

False Explanation: 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.

The probability density function of a continuous uniform distribution is positive for all values between -infinity and +infinity

False Explanation: The uniform distribution is defined on a bounded interval, denoted by [a, b].

The probability density function of a continuous random variable is the counterpart to the probability mass function of a discrete random variable.

True

A Type 1 error is committed when we reject the null hypothesis which is actually true.

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

A continuous random variable is characterized by uncountable values and can take on any value within an interval

True Explanation: A continuous random variable is characterized by infinitely uncountable values and can take on any value within an interval.

A hypothesis test regarding the population mean is based on the sampling distribution of the sample mean.

True Explanation: A hypothesis test regarding the population mean µ is based on the sampling distribution of the sample mean X−.

The standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one

True Explanation: A standard normal distribution is a special case of the normal distribution with a mean equal to zero and standard deviation equal to one.

For any population proportion p, the sampling distribution of (P with a bar on top) will be approximately normal if the sample size n is sufficiently large. As a general guideline, the normal distribution approximation is justified when np ≥ 5 and n(1 −p) ≥ 5.

True Explanation: As a general guideline, the normal distribution approximation for any population proportion is justified when np ≥ 5 and n(1 −p) ≥ 5

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 Explanation: As a general guideline, we use the alternative hypothesis as a vehicle to establish something new, that is, to contest the status quo. In general, the null hypothesis regarding a particular population parameter of interest is specified with one of the following signs: =, ≥, ≤; the alternative hypothesis is then specified with the corresponding opposite sign: ≠, >, <.

For a given confidence level 100 ( 1− α )% and population standard deviation σ, the width of the confidence interval for the population mean is wider, the smaller the sample size n.

True Explanation: For a given confidence level 100 ( 1− α ) % and sample size n, the width of the confidence interval for the population mean is wider, the greater the population standard deviation σ.

If a random sample of size n is taken from a normal population with a finite variance, then the statistic formula finding T follows the t distribution with n-1 degrees of freedom

True Explanation: If a random sample of size n is taken from a normal population with a finite variance, then the statistic T = X¯¯¯− μ S / n √ follows the tdf distribution with (n −1) degrees of freedom, df.

For a given sample size, any attempt to reduce the likelihood of making one type of error (Type 1 of Type 2) will increase the likelihood of the other error.

True Explanation: 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.

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

True Explanation: Most statistical methods presume simple random sample.

Nonresponse bias occurs when those responding to a survey or poll differ systematically from the non-respondents

True Explanation: Nonresponse bias occurs when those responding to a survey or poll differ systematically from the nonrespondents.

The probability density function of a continuous random variable can be regarded as a

True Explanation: The area under any probability density function is 1, and as for any discrete random variable X with values x1, x2, x3, . . . , xn , ΣP(X=xi)=1 .

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

True Explanation: The estimate will be less precise if the variability of the underlying population is high or a small segment of the population is sampled.

The exponential distribution is related to the Poisson distribution.

True Explanation: The exponential distribution is related to the Poisson distribution even though the Poisson distribution deals with discrete random variables.

The main ingredient for developing a confidence interval is the sampling distribution of the underlying statistic.

True Explanation: The main ingredient for developing a confidence interval is the sampling distribution of the underlying statistic.

The mean of a continuous uniform distribution is simply the average of the upper and lower limits of the interval on which the distribution is defined

True Explanation: The mean or the expected value for the continuous uniform distribution defined as E(X)=μ=a+b2 , where a and b are lower and upper limits of values.

Under the assumption that the null hypothesis is true as a equality, the p-value is the likelihood of observing a sample mean that is at least as extreme as the one derived from the given sample.

True Explanation: The p-value is the likelihood of obtaining a sample mean that is at least as extreme as the one derived from the given sample, under the assumption that the null hypothesis is true as an equality.

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

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

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

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