Statistics Exam 3 Definitions (Ch 7, 8, 9, 10)
The points at x=_______ and x=_______ are the inflection points on the normal curve.
The points are x=μ−σ and x=μ+σ.
What happens to the probability of making a Type II error, β, as the level of significance, α, decreases? Why?
The probability increases. Type I and Type II errors are inversely related.
Explain why the t-distribution has less spread as the number of degrees of freedom increases.
because, as n increases, s becomes closer to σ by the law of large numbers.
A Type II error
occurs if we fail to reject the null hypothesis, even though it is actually false.
A Type I error
occurs if we reject the null hypothesis, even though it is actually true.
Suppose the proportion of a population that has a certain characteristic is 0.05. The mean of the sampling distribution of p hat from this population is
.05
As the number of samples increases, the proportion of 95% confidence intervals that include the population proportion approaches ______.
.95
When constructing 95% confidence intervals for the mean when the parent population is right skewed and the sample size is small, the proportion of intervals that include the population mean approaches _____ as the sample size, n, increases.
.95
If a 95% confidence interval results in a sample proportion that does not include the population proportion, then the sample proportion is more than ______ standard errors from the population proportion.
1.96
For the shape of the distribution of the sample proportion to be approximately normal, it is required that np(1−p)≥______.
10
What percent of sample proportions results in a 98% confidence interval that does not include the population proportion?
2%
What percent of sample proportions results in a 98% confidence interval that includes the population proportion?
98%
The Empirical Rule
Approximately 68% of the area under the normal curve lies within 1 standard deviation of the mean Approximately 95% w/in 2 std dev Approximately 99.7% w/in 3 std dev
What do you conclude about the impact of large samples on the P-value?
As n increases, the likelihood of rejecting the null hypothesis increases. However, large samples tend to overemphasize practically insignificant differences
Explain what a P-value is. Choose the correct answer below.
A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true.
How does the decrease in confidence affect the sample size required?
Decreasing the confidence level decreases the sample size needed.
Determine if the following statement is true or false. When testing a hypothesis using the P-value Approach, if the P-value is large, reject the null hypothesis.
False
Determine whether the following statement is true or false. Sample evidence can prove that a null hypothesis is true.
False
Determine whether the following statement is true or false. To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
False
True or False: The population proportion and sample proportion always have the same value.
False
What is the criterion for rejecting the null hypothesis using the P-value approach? Choose the correct answer below.
If P-value < α, reject the null hypothesis.
What is at the "heart" of hypothesis testing in statistics?
Make an assumption about reality, and collect sample evidence to determine whether it contradicts the assumption.
_____________________ refers to the idea that although small differences between the statistic and parameter stated in the null hypothesis are statistically significant, the difference may not be large enough to cause concern or be considered important.
Practical significance
Explain what "statistical significance" means.
Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true.
Explain the difference between statistical significance and practical significance.
Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application.
Suppose a simple random sample of size n is obtained from a population whose distribution is skewed right. As the sample size n increases, what happens to the shape of the distribution of the sample mean?
The distribution becomes approximately normal.
What happens to the graph of the normal curve as the standard deviation decreases?
The graph of the normal curve compresses and becomes steeper.
What happens to the graph of the normal curve as the mean increases?
The graph of the normal curve slides right.
As the sample size n increases, what happens to the standard error of the mean?
The standard error of the mean decreases
Suppose that X is a binomial random variable. To approximate P(3≤X<7) using the normal probability distribution, we compute P(3.5≤X<7.5). Is the statement below true or false?
The statement is false.
The area under the standard normal curve to the left of z=5.30 is 1.
The statement is false. The total area under the standard normal curve is 1, and there is some area to the right of z=5.30, so the area to the left of z=5.30 must be less than 1.
A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal. Is this statement true or false?
The statement is true.
The normal curve is symmetric about its mean, μ.
The statement is true. The normal curve is a symmetric distribution with one peak, which means the mean, median, and mode are all equal. Therefore, the normal curve is symmetric about the mean, μ.
The mean of the sampling distribution of p hat is p. Is the statement below true or false?
True
The _______ _______ is a statement we are trying to find evidence to support.
alternative hypothesis
Why does the margin of error decrease as the sample size n increases?
because the difference between the statistic and the parameter decreases. This is a consequence of the Law of Large Numbers.
Why does the margin of error increase as the level of confidence increases?
because the larger the expected proportion of intervals that will contain the parameter, the larger the margin of error.
When constructing 95% confidence intervals for the mean when the parent population is right skewed and the sample size is small, the proportion of intervals that include the population mean is (above, below, equal to) 0.95.
below
A _______________ for an unknown parameter consists of an interval of numbers based on a point estimate.
confidence interval
What effect does increasing the sample size have on the probability? Provide an explanation for this result.
decreases the probability because σx bar decreases as n increases.
How does increasing the level of confidence in the estimate affect sample size?
increases the sample size required. For a fixed margin of error, greater confidence can be achieved with a larger sample size.
If the normality requirement is not satisfied (that is, np(1minus−p) is not at least 10), then a 95% confidence interval about the population proportion will include the population proportion in ________ 95% of the intervals
less than
The ________________ represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained.
level of confidence
The _____________ , α,is the probability of making a Type I error.
level of significance
A continuous random variable is __________, or has a _______________, if its relative frequency histogram has the shape of a normal curve.
normally distributed normal probability distribution
The __________ hypothesis, denoted Upper H 0H0, is a statement to be tested, and is a statement of no change, no effect, or no difference.
null
A ______________ is the value of a statistic that estimates the value of a parameter.
point estimate
A ________ ________ is the value of a statistic that estimates the value of a parameter.
point estimate
The _____ _____, denoted p hat, is given by the formula p hat =_____, where x is the number of individuals with a specified characteristic in a sample of n individuals.
sample proportion, x/n
The standard deviation of the sampling distribution of x bar, denoted σx bar, is called the _____ _____ of the _____.
standard error of the mean.
When observed results are unlikely under the assumption that the null hypothesis is true, we say that the result is _____________ and we reject the null hypothesis.
statistically significant
Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sample distribution of x bar has mean ux bar =______ and standard deviation σx bar equals=______.
μx bar = μ σx bar = σ/square root of n.