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Final Exam
Introduction to Statistics- Chapters 6 and 7
| Question | Answer |
|---|---|
| Point Estimate definition | A single value estimate for a population parameter |
| What is the most unbiased point estimate of the population mean (mu)? | The sample mean (x bar) |
| What does the validity of an estimation method do when you use a sample statistic that is unbiased an has low variability? | It increases |
| The mean of all possible samples means of the same size _________ (equals, greater than, less than) the population mean | equals |
| x bar is a ______ (biased/unbiased) estimator of mu | unbiased |
| What does sigma/square root of n mean (in words)? | The standard error |
| What happens to the standard error of a sample mean when n increases? Does it become more or less variable? | It decreases; less variable |
| Interval estimate definition | Is an interval used to estimate a population parameter |
| What is an interval? | A range of values |
| Level of confidence definition | The probability that the interval estimate contains the population parameter, assuming that the estimation process is repeated a large number of times. |
| When n is greater or equal to 30, what does the sample distribution of sample means approximate? | A normal distribution |
| The level of confidence c is what area on a standard normal curve? | The area under the standard normal curve between the critical values, -z and z |
| Critical values definition | The values that separate sample statistics that are probable from sample statistics that are improbable/unusual |
| What is the formula for the area under one tail? | 1/2 (1-c) |
| Sampling error definition | The difference between the point estimate and the actual parameter value |
| When mu is estimated, what is the sampling error? | The difference of x bar - mu |
| In most cases mu is _____ and x bar _____ from sample to sample | unknown; varies |
| What is another way to say margin of error E? | The maximum error estimate or error tolerance |
| What is the margin of error, given a level of confidence? | The greatest possible distance between the point estimate and the value of the parameter it is estimating |
| What is the formula for the margin of error for mu when sigma is known? | E= z sub c (sigma sub x bar)= z sub c (sigma/square root of n) |
| What conditions must be met to use the margin of error formula? | The sample is random At least one of the following is true: the population is normally distributed or n is greater than or equal to 30 |
| How to construct a confidence internal of a population parameter? | Using a point estimate and a margin of error, you can construct an interval estimate (confidence interval) of a population parameter such as mu |
| A c-confidence interval for a population mean mu is... (formula) | x bar - E is less than mu which is less than x bar + E |
| A c-confidence interval for a population mean mu is... (words) | The probability that the confidence interval contains mu is c, assuming that the estimation process is repeated a large number of times |
| what are the steps to constructing a confidence interval for a population mean when sigma is known | Verify that sigma is known/sample is random/population is normally distributed or n is > equal to 30 Find the sample statistics n and x bar, critical values z sub c that corresponds to the given level of confidence, margin of error E, and endpoints |
| As the level of confidence increases, what does the confidence interval do? What does the precision of the estimate do? | Widens; decreases |
| How do you improve the precision of an estimate without decreasing the level of confidence? | Increase sample size |
| When do you use the margin of error formula? | To find how large a sample size is needed to guarantee a certain level of confidence for a given margin of error |
| Given c-confidence level and a margin of error E, the minimum ample size n need to estimate the population mean mu is ... (formula) | n= (z sub c time sigma/E)^2 If n is not a whole number than round to next whole number When sigma is unknown you can estimate it using s, provided that there are at least 30 members |
| What are the properties of the t-distribution? | The mean, median, and mode are 0 It is bell-shaped and symmetric about the mean The total area under it's curve is equal to 1 The tails under the t-distribution are "thicker" than those in the standard normal distribution The standard deviation is > 1 |
| What does the sample distribution of x bar follow a t-distribution? | When it does not follow a normal distribution |
| If the distribution of a random variable x is approximately normal, the... (formula) | t= x bar- mu/ (s/square root of n) |
| What are degrees of freedom? | The number of free choices left after a sample statistic as x bar is calculated |
| When using the t-distribution to estimate a population mean, how do you calculate the degrees of freedom? | one less than the sample size |
| What happens to a t-distribution as the degrees of freedom increases? | The t-distribution approaches the standard normal curve |
| How to calculate a confidence interval for mu when sigma is not known? | E= t sub c (s/square root n) |
| What must you verify before using the margin of error for mu formula when sigma is unknown? | the population is normally distributed/n is greater than or equal to 30 The sample is random |
| Steps to constructing a confidence interval for a population mean | Verify sigma is not known, the sample is random, and population mean is normally distributed/n is greater than or equal to 30 Find n, x bar, and s Identify: df, c, and t sub c Find the margin of error Find endpoints and form confidence interval |
| Hypothesis Test defintion | A process that uses sample statistics to test a claim about the value of a population parameter |
| How do you test a population parameter? | You carefully state a pair of hypotheses test |
| What is a statement about a population parameter called? | A statistical hypothesis |
| What are the two types of hypothesis? | Null and alternative |
| What is a null hypothesis? | A statistical hypothesis that contains a statement of equality |
| What is an alternative hypothesis? | The complement of the null hypothesis; contains a statement of strict inequality |
| When preforming a hypothesis test what are the two decisions you can make? | Reject the null hypothesis or fail to reject the null hypothesis |
| What are the two types of errors you can make? | Type I and Type II |
| What is a type I error? | When you reject the null hypothesis when it is actually true |
| What is a type II error? | When you do not reject the null hypothesis when it is false |
| What is the level of significance? | The maximum allowable probability of making a type I error |
| What is the sample statistic called? | Test statistic |
| What is the test statistic converted into with the assumption that the null hypothesis is true | The standardized test statistic |
| If given the population parameter mu, what do you use for the test statistic and the standardized test statistic? | Test statistic: x bar Standardized test statistic: z (sigma known), t (sigma unknown) |
| If given the population parameter p, what do you use for the test statistic and the standardized test statistic? | Test statistic: p^ Standardized test statistic: z |
| If given the population parameter sigma squared, what do you use for the test statistic and the standardized test statistic? | Test statistic: s squared Standardized test statistic: X squared |
| What are the three types of hypothesis tests? | Left tailed Right tailed Two tailed |
| How to know what type of hypothesis test you should use? | Left tailed: < Right tailed: > Two tailed: not equal to |
| What is the P-value of a hypothesis test? | The probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data |
| What does a smaller P-value provide evidence for? | A smaller P-value provides more evidence to reject the null hypothesis, but it is not proof that the null hypothesis is false |
| What is the decision rule? | A way to decide to reject the null hypothesis or fail to reject the null hypothesis |
| If P is less than or equal to a, you should ___ the null hypothesis | reject |
| If P is greater than a, you should ___ the null hypothesis | fail to reject |
| Does failing to reject the null hypothesis mean that you have accepted it? | No, it means there is not enough evidence to reject the null hypothesis |
| What are the steps for Hypothesis testing? | State claim. Identify null/alternative hypothesis Specify level of significance Determine standardized sampling distribution Calculate test statistic/its standardized test statistic Find the P-value Interpret the decision in context of the claim |
| How do we label the probability of accepting a false null hypothesis? Is this a type I or Type II error? | Type II error |
| Suppose we perform a hypothesis testing at 4% level of significance. Give a p-value that would indicate that we should fail reject the null hypothesis. | p=.05, any p-value >.04 will work |
| Describe the differences between the Standard Normal Distribution and the Studentβs t- distribution | t-distribution have fatter tails and shallower peak than a standard normal There is a single standard normal distribution but many studentβs t distributions |
| Under what circumstances should we use the Standard Normal Distribution? | π is known or π β₯ ππ |
| Under what circumstances should we use the Studentβs t-distribution | π is unknown and n < 30 |
| What are the two types of hypotheses used in a hypothesis test? How are they related? | Null hypothesis π―π and Alternative hypothesis π―π or π―π. Both have the same parameter and value, but π―π contains the equality and π―π is the complement. |
| What are the two decisions that you can make from performing a hypothesis test? | Reject π―π or fail to reject π―π |
| Does failing to reject the null hypothesis mean that the null hypothesis is true? Explain. | No. It could be type II error or the answer may change based on a |
| You are designing a statistical test and decide to lower the significance level (πΌ) from 0.05 to 0.01 reduce the risk of a "False Alarm." What happens to the probability of making a Type II error as a result of this decision? | Increases |
| A factory has an automated alarm system. The null hypothesis is that the machinery is operating safely. How would you classify the error if the alarm goes off and shuts down the factory, but the machinery was actually operating perfectly safely? | Type I error |
| An email spam filter is designed to catch junk mail. The null hypothesis is that an incoming email is "legitimate" (not spam). How would you classify the error if a malicious scam email bypasses the filter and lands in your inbox? | Type II Error |
| A two-tailed hypothesis test is preformed with a corresponding 95% confidence interval. If the hypothesized mean π falls outside the confidence interval, what conclusion should be reached regarding the null hypothesis at the 0.05 significance level? | Reject π―π |
| What variable equals margin of error? | E equation |
| What variable equals sample mean? | x bar |
| What does x bar stand for? | the sample mean |
| What does mu stand for? | population mean |
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