Stats Mid-term 1

T or F Another name for the standard deviation of the sample mean is the "standard error of the mean.

True

Theoretical sampling distribution of the mean

displays all the possible sample means along with their classical probabilities

discrete uniform probability

each event has the same probability

the probability of a confidence interval is a complement to the

significance level i.e. confidence level is 95% so the significance level is 5%

increasing your confidence level has what effect on the true pop mean?

the interval estimate of the true population mean becomes wider and less precise.

degrees of freedom

the number of values that are free to be varied given the info, such as the sample mean, is known

dependent samples

the observation from one sample is related to an observation from another sample

p-value (aka observed level of significance)

the smallest level of significance at which the null hypothesis will be rejected, assuming the null hypothesis is true

T or F If the population distribution is unknown, in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

True

T or F Suppose μ= 50 and σ2 = 100 for a population. In a sample where n = 100 is randomly taken, 95% of all possible sample means will fall between 48.04 and 51.96

True

T or F The Central Limit Theorem ensures that the sampling distribution of the sample mean approaches normal as the sample size increases.

True

T or F The amount of tea leaves in a can from a particular production line is desired to be 100 grams and σ is known to be 6 grams. A sample of 36 cans is to be selected and a sample mean of 105 grams is observed. One can conclude that this machine is out of adjustment.

True

T or F The standard error of the mean is also known as the standard deviation of the sampling distribution of the sample mean.

True

T or F Let the desired error (Xbar - μ) be 4, the std. deviation of the population be 20, and the confidence level 95%. The sample size that yields this error rate is about 97.

True, (Z*σ/e)^2 = n, the sample size that gives us the error rate. Here we have (1.96*20/4)^2 = 9.8^2 = 96.04. We round up to n=97

T or F If σ = 10 and n = 64, we would expect the standard error of the sample mean to be about 1.25.

True, 10/(64^0.5) = 10/8 = 1.25

T or F Placing the burden of proving guilt on the prosecution in the US justice system can be interpreted as the a desire to limit type-I errors.

True, A type I error would mean declaring an innocent person guilty. The burden of proof on the prosecution can be seen as an attempt to lower the likelihood of a type I error.

T or F If IQ is normally distributed with a mean of 100 and a standard deviation of 15, then about 68 percent of the population should lie within the range of 85 to 115.

True, About 68% of observations fall within ± 1 standard deviation from the mean

T or F E(aX+b) = a E(X)+ b, but V(aX+b) = a2 V(X).

True, Adding a constant does not change variance, but multiplying the X value by a scalar does.

T or F The expected value of the median is equal to the true population median, but because the variance of the median is substantially higher than the variance of the sample mean, the median is a "less efficient" estimator of the true population mean.

True, An estimator with a higher variation is said to be less "efficient."

T or F Point estimates, such as the sample mean or sample proportion give us very little indication as to how confident we should be in that estimate.

True, Confidence intervals provide us with more information about the range of values that the true population parameter might take on.

T or F One way of executing a hypothesis test of the two-tail variety is simply to form the confidence interval and see whether the hypothesized value falls within that confidence interval.

True, If the hypothesized values doesn't fall within the confidence interval, we reject the null.

T or F We control the level of type-I errors quite handily. The problem is that when we lower this type of error we magnify the likelihood of a type-II error.

True, If we lower the a level of significance we raise the beta error probability.

T or F If the true proportion is 0.4 and the sample size is 96, we would expect the standard error of the sample proportion of be 0.05

True, Standard error of the proportion is (.4*.6/96)^0.5 = (.24/96)^0.5 = .0025^0.5 = .05.

T or F The standard error of the sample mean is the standard of the population divided by the square-root of the sample size.

True, That's how we compute it.

T of F If the population average gasoline purchase is for 10.4 gallons and the variance is known to be 16 gallons, then in a random sample of 36 purchases the standard error of the mean would be 2/3.

True, The standard error is sqrt(16/36) = 4/6 = 2/3

T or F The Point Estimate ± (Critical Value)(Standard Error) is the general formula for a confidence interval.

True, This is a general formula for a confidence interval. The critical value is the Z or t value associated with the level of confidence and degrees of freedom. The standard error depends upon the test statistic and sample size. Usually it is the standard deviation estimate divided by the square root of n.

T or F The probability of accepting a false null hypothesis is called a type II error or Beta error

True, This is the definition of a Beta error. In our test, we fall within the acceptance range, and do not reject the null. If the null is not true, we have committed a Type II or Beta error.

T or F The Finite Population Correction Factor (FPC Factor) should be employed whenever the sample size is about 5 percent or more of the population.

True, This is the general rule. 0.95^2 = 0.975, so at the 5% sample of the parent populations size yields a sample standard error that is about 2.5% lower.

T or F It takes a rather large sample size for the sample proportion to be normally distributed especially when the true proportion of successes is low.

True, This is what was revealed by the MRunsProportion simulations.

T or F The sample variance is a biased estimate of the true population variance when we divide the sum of squared deviations from the sample mean by N instead of N-1.

True, We need to divide the sum of squared deviations from the sample mean by n-1 to have an unbiased estimator of population variance.

T or F A random sample of 50 provides a sample mean of 31 with a standard deviation of S=14. The upper bound of a 90% confidence interval estimate of the population mean is 33.32.

False

T or F If the amount of gasoline purchased per car at a large service station has a population mean of $15 and a population standard deviation of $4 and a random sample of 4 cars is selected, there is approximately a 68.26% chance that the sample mean will be between $13 and $17.

False

T or F The t-distribution is the limit of the normal distribution as the number of degrees of freedom approaches infinity.

False, No, the normal distribution is the limit of the t distribution.

T or F To reduce the standard error of the mean by 50 %, one would need to double the sample size

False, Say std dev = 5. With n=25, std. error of mean would be 1. To reduce it to 0.5, n would have to equal 100. We have to quadruple the sample size.

T or F The sampling distribution of the sample mean is normally distributed regardless of sample size.

False, it can be t-distributed with small sample sizes.

T of F If the population distribution is skewed, in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

True

T or F A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

True

T or F Increasing the sample size is one way of reducing type-II error potentials.

True

confidence interval

a range of values used to estimate a population parameter and is associated with a specific confidence level.

point estimator

a single value that best describes the population of interest, the sample mean being the most common. advantage is it's easy to calculate, but disadvantage is how accurate the estimate is

Standard error of the mean

aka the standard deviation of the sample means. As the sample size increases, the standard error/standard deviation of the means becomes smaller. This is denoted by sigma of x bar= (sigma/sqrt of n)

hypothesis

an assumption about a population parameter

How do alpha and beta values interact with each other?

ideally, we want smaller alpha and beta results. - for a given sample size, reducing the value of alpha will result in an increase in the value of beta. - the only way to reduce both alpha and beta simultaneously is to increase the sample size. - once the sample size=the population size, the values of alpha and beta = 0. (that's not a recommended strategy, though)

How can you reduce the width of your confidence interval while maintaining the same confidence level?

increase sample size N

one tail hypothesis test

involves the alternative hypothesis being sated as < or >, there's only 1 rejection region

level of significance

is the probability of making a type 1 error.

null hypothesis

null hypothesis refers to a general or default position: that there is no relationship between two measured phenomena,[1] or that a potential medical treatment has no effect.

type 1 error (alpha error)

occurs when the null hypothesis is not accepted when in reality it is true

type 2 error (beta error)

occurs when we fail to reject the null hypothesis when in reality it is not true

interval estimate

provides a range of values that best describes the population, this helps us deal with the uncertainty provided in the point estimator.

sampling distribution of the mean

refers to the pattern of sample means that will occur as samples are drawn from the population at large

T or F With a sample size of 64 a sample mean of 12 is computed and the standard deviation is estimated at 4.0. We can be more than 95% confident that the true mean will be between 11 and 13.

True

properties of a student's t-distribution

1. bell shaped and symmetrical around the mean 2. the shape of the curve depends on the degrees of freedom which, when dealing with the sample mean, would be equal to n-1 3. the area under the curve is equal to 1 4. the t-distribution is flatter than the normal distribution, but as degrees of freedom increases, the shape of the t-distribution approaches the normal distribution.

what conditions must be met to use the t-distribution

1. the population follows the normal (or approximately the normal) distribution 2. the sample size is less than 30 3. the population standard deviation, sigma, is unknown and must be approximated by s, the sample standard deviation

T or F In reference to the previous problem, we select a 5% level of significance. a sample of 400 is taken and the sample proportion preferring Cola X is 0.16. After performing the computations, we accept the null hypothesis. There is insufficient evidence to conclude that the true proportion has fallen.

False

T or F The t-distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

False

T or F A race car driver tested his car for time from 0 to 60 mph, and in 20 tests obtained an average of 4.85 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 time is 4.52 seconds to 5.18 seconds.

False

T or F For a very large sample, say, 500, we can be 99% confident that the true mean falls within ± 2.58 standard error of one sample mean we obtain from our costly survey.

False, The true mean falls within a specified confidence interval or it doesn't. It is a parameter. It doesn't vary. In repeated samples, however, 99% of the sample confidence intervals we construct would include the true population mean.

z test or t test?

If the test statistic follows a Student's t distribution in the null hypothesis - which is common where the underlying variable follows a normal distribution with unknown scaling factor, then the test is referred to as a one-tailed or two-tailed t-test. If the test is performed using the actual population mean and variance, rather than an estimate from a sample, it would be called a one-tailed or two-tailed or Z-test.

how large must a sample size n be to construct a confidence interval?

N>=30

T or F A sample of 100 fuses from a very large shipment is found to have 10 that are defective. The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0.0422 and 0.1588.

True

T or F A sample size of 5 provides a sample mean of 9.6. If the population variance is known to be 5 and the population distribution is assumed to be normal, the lower limit for a 90% confidence interval is 7.96.

True

T or F For distributions such as the normal distribution, the arithmetic mean is considered more stable from sample to sample than the median. This property is known as "efficiency."

True

T or F Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

True

T or F Past studies have revealed that Cola X has a 20 percent preference rating among consumers. There is suspicion that the market share has declined. The null hypothesis is that the true proportion is 0.2 and the alternative is that the true proportion is less than 0.2.

True

T or F The difference between the upper limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

True

T or F The sampling error can either be positive or negative.

True

T or F The t-distribution approaches the standardized normal distribution when the number of degrees of freedom increases. Correct Response A) True

True

T or F The t-distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

True

central limit theorem

as the sample size, n, gets larger, the sample means tend to follow a normal probability distribution and tend to cluster around the true population mean. This holds true regardless of the distribution of the population from which the sample was drawn.

proportion data follows what type of distribution?

binomial, but the binomial can be approximated by the normal distribution under the following conditions: np>=5 & nq>=5

pooled estimate of the standard deviation

combines two sample variances into one variance

parameter

data that describes characteristics about a population, it's a numerical descriptor (i.e. the mean)

sampling distribution for the difference in means

describes the probability of observing various intervals for the difference between two sample means

standard error of the difference between two means

describes the variation in the difference between two sample means

margin of error "E"

determines the width of the confidence interval

alternative hypothesis

the opposite of the null, this is generally a research hypothesis or claim

confidence level

the probability that the interval estimate will include the population parameter, such as the mean

standard error of the proportion

the standard deviation of the sampling proportions

independent samples

there is no relationship in the observations between the sample

what is the purpose of a hypothesis test

to verify the validity of a claim about a population based on a single sample

two tail hypothesis test

used whenever the alternative hypothesis is expressed as "not equal."