Ch. 5 Normal Distribution
For the first group of questions - use your half sheet on the 68-95-99.7 Rule (Empirical Rule).
A new group of IQ tests are standardized to a Normal Model, N(110,14).
Answer:
What percent of the data values fall between the IQ's of 96 and 124?
Using the Normal Distribution from #1:
How many standard deviations from the mean is the top 16% of the IQ scores?
What is the IQ score needed to be in the top 16%?
What percentile would this be?
Hint: remember, percentiles are the percent to the LEFT of a given point
Then shade the corresponding area under the normal curve in the 'show your work' area below.
Using the Normal Distribution from #1:
How many standard deviations is the top 2.5% from the mean?
What is the IQ score needed to be in the top 2.5%?
What percentile would this be?
Hint: remember, percentiles are the percent to the LEFT of a given point
Then shade the corresponding area under the normal curve in the 'show your work' area below,.
Using the Normal Distribution from #1:
What percent of the IQ scores are 82 or lower?
What percent of the IQ scores are 124 or lower?
What percent of the IQ scores are 68 or lower?
Hint: remember, percentiles are the percent to the LEFT of a given point
Fill in the blanks:
Using the normal distribution model:
N(3.25, 1.4) Mean=
What does the z-score mean?
Suppose that a Normal model described student scores in a history class.
Francisco has a standardized score (z-score) of +2.5.
This means that Francisco’s score...
If the heights of 6th graders at Danville Middle School follow a normal distribution with the mean height of 58.5 inches and the standard deviation of 2.45 inches, give the notation for the Normal model.
Use the format: N(mean, std dev)
The Delorean speeds from 'Back To The Future' follow a Normal Distribution, with the model N(80, 7.7).
What percent of the runs will give the Delorean a speed less than 68.45 mph?
First: what is the z-score for 68.45?
Use the z-score formula:
Second: answer,
what is the proportion of runs that will give the Delorean a speed less than 68.45 mph?
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
What percent of the runs will give the Delorean a speed greater than 85 mph?
First: what is the z-score for the speed?
Remember to round to two places.
Second: be careful, I am asking for the percent GREATER than 85 mph, think about what you need to do, collaborate!
What is the proportion of speeds less than 85 mph?
Third: What do you need to do with the value to find the proportion higher?
Add, subtract, multiply or divide?
What percent of the runs will give the Delorean a speed greater than 85 mph?
Hint: the area to the left and the right need to add to 1.0, the total area under the curve.
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
What percent of the runs will give the Delorean a speed between 70 and 95 mph?
Remember it follows the model: N(80, 7.7).
First: what is the z-score for the speed of 70 mph?
Remember to round to two places.
what is the z-score for the speed of 95 mph?
Second: what proportion corresponds to 70 mph?
what proportion corresponds to 95 mph?
Third: answer, what will you do with the two proportions?
Add, subtract, multiply or divide?
What percent of the runs will give the Delorean a speed between 70 and 95 mph?
Hint: you want the area BETWEEN the two points, you will need to subtract!
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
High levels of cholesterol in the blood increase the risk of heart disease. For teenage boys, the distribution of blood cholesterol is approximately normal with mean μ = 151.6 milligrams of cholesterol per deciliter of blood (mg/dl) and standard deviation σ = 25 mg/dl.
What is the Normal model for this situation?
Use the format: N(mean, std. dev.)
What proportion of teen boys have cholesterol levels less than 100 mg/dl?
First: what is the z-score for 100 mg/dl?
Second: What is the proportion?
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
Cholesterol levels of 200 or higher are considered high for teenagers.
What percent of teen boys have high cholesterol?
First: what is the z-score for 200 mg/dl?
Second: What is the proportion to the left of the z-score?
What do you need to do with the value to find the proportion higher?
Add, subtract, multiply or divide?
Third: what is the proportion GREATER than 200 mg/dl?
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
Cholesterol levels between 170 mg/dl and 200 mg/dl are considered borderline high for teenagers. What percent of teen boys have borderline high cholesterol levels?
First: what is the z-score for 170 mg/dl?
what is the z-score for 200 mg/dl?
Second: What is the proportion to the left of 170?
What is the proportion to the left of 200?
Third: what will you do with the two proportions? Add, subtract, multiply or divide?
what percent of teen boys have borderline high cholesterol levels?
THEN: Shade in the probability area that you are calculating in the 'show your work' area.
People with z-scores greater than 2.5 on an IQ test are considered as geniuses.
IQ tests have a score of a mean of 100 and SD of 16 points.
What is the cut off score to show someone is a genius? Hint: use the z-score.
Chapter 6 Sampling Distributions
A study of voting chose 663 registered voters at random shortly after an election. Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election.
Match up the following statements about these percentages?
| Stavka koja se može prevući | arrow_right_alt | Odgovarajuća stavka |
|---|---|---|
663 regsitered voters | arrow_right_alt | Parameter |
56% | arrow_right_alt | Statistic |
All voters | arrow_right_alt | Sample |
72% | arrow_right_alt | Population |
Vermont is particularly beautiful in early October when the leaves begin to change color. At that time of year, a large proportion of cars on Interstate 91 near Brattleboro have out-of-state license plates.
Suppose a Vermont state trooper randomly selects 50 cars driving past Exit 2 on I-91, records the state identified on the license plate, and calculates the proportion of cars with out-of-state plates.
What is the Vermont state trouper recording?
Which of the following describes the sampling distribution of the sample proportion in this context?
A polling organization wants to estimate the proportion of voters who favor a new law banning smoking in public buildings.
The organization decides to increase the size of its random sample of voters from about 1500 people to about 4000 people right before an election.
The effect of this is to
The central limit theorem is important in statistics because it allows us to use the normal distribution to find probabilities involving the sample mean if the
At a high school, 85% of students are right-handed.
Let X = the number of students who are right-handed in a random sample of 10 students from the school.
What is the sample size? n=
What is the mean of the sampling distribution?
What is the standard deviation of the sampling distribution?
Scores on the mathematics part of the SAT exam in a recent year followed a normal distribution with mean 515 and standard deviation 114.
You choose an SRS of 100 students and calculate mean SAT Math score.
Match up the following for this situation.
| Stavka koja se može prevući | arrow_right_alt | Odgovarajuća stavka |
|---|---|---|
11.4 | arrow_right_alt | sample size (n) |
100 | arrow_right_alt | |
515 | arrow_right_alt | |
114 | arrow_right_alt |
In a congressional district, 52% of the registered voters are Democrats. An SRS of 100 voters is going to be polled.
What is the mean of the sampling distribution of the sample proportion of democrat voters?
What is the standard deviation of the sampling distribution of the sample proportion of democrat voters?
What is the probability of getting less than 50% Democrats in a random sample of size 100?
Keep your answer as the decimal proportion, keep all decimal places (there will be four)
Use the information in the previous question.
What is the probability of getting a sample of 100 registered voters with more than 65% democrats?
Keep all decimals.
Using the information above, suppose an actual sample of 100 of the registered voters actually resulted in 65% democrats. Would you have convincing evidence that the estimated 52% democrat voters was incorrect? Explain.
The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $48 and the standard deviation is $20.
The distribution is not normal: Many households pay a base rate for low-speed access, but some pay much more for faster connections so the distribution is strongly right skewed.
A sample survey asks an SRS of 500 households with Internet access how much they pay per month. Let X-bar be the mean amount paid by the members of the sample.
Calculate the mean and standard deviation of the sampling distribution of the sample means.
Sampling distribution sample mean =
Sampling distribution of sample means standard deviation =
Round to three places past the decimal.
Use the information from the previous question:
What is the shape of the population and the sampling distribution of x-bar ? Justify.
Use the information in the previous two questions:
In a sampling distribution, find the probability that the average amount paid by the sample of households exceeds $50.
z-score =
Probability =
Use the information from the previous 3 questions:
In a sampling distribution, what is the probability that the average amount paid by the sample of households is less than $46.50.
z-score =
Probability =
Chapter 7: Confidence Intervals
Tim purchased a random sample of clementines at a local grocery store. The 95% confidence interval for the mean weight of all clementines at this store is 76.6 grams to 90.1 grams. Interpret this confidence interval.
In a recent year, 73% of first-year college students identified “being very well-off financially” as an important personal goal.
A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important.
Construct a 95% confidence interval for the true proportion of all first-year students at the university who would identify being well-off as an important personal goal.
Check your formula chart, round to three places past the decimal.
Lower bound:
Upper bound:
Based on your previous answer:
Explain what the confidence interval tells you about whether the national value of 73% holds at this university.
Melissa and Madeline love pepperoni pizza, but sometimes they are disappointed with the small number of pepperonis on their pizza.
To investigate, they went to their favorite pizza restaurant at 10 random times during the week and ordered a large pepperoni pizza. Here are the number of pepperonis on each pizza.
47 36 25 37 46 36 49 32 32 34
Statsmedic.com gives an average of 37.4 pepperonis and a Sx= 7.662
Construct and interpret a 95% confidence interval for the true mean number of pepperonis on a large pizza at this restaurant.
Check your formula chart, round to three places past the decimal.
Lower bound:
Upper bound:
Based on your previous answer:
Explain what the confidence interval tells you about whether the manager's requirement of 40 pepperonis on each large pizza is being followed.
The puzzle editor of a game magazine asked 43 randomly selected subscribers how long it took them to complete a certain crossword puzzle. The 99% confidence interval for the median completion time for all subscribers is 15.2 to 18.6 minutes.
Explain what would happen to the length of the interval if the sample size were increased to 200 students.
The puzzle editor of a game magazine asked 43 randomly selected subscribers how long it took them to complete a certain crossword puzzle. The 99% confidence interval for the median completion time for all subscribers is 15.2 to 18.6 minutes.
Explain what would happen to the length of the interval if the confidence level were increased to 99%.
Chapter 8
Identify the types of Errors when performing a significance test:
Ha is TRUE and you REJECT Ho
Ho is TRUE and you REJECT Ho
Ho is TRUE and you FAIL TO REJECT Ho
Ha is TRUE and you FAIL TO REJECT Ho
Type I Error
Type II Error
No Error
Flu vaccine A drug company has developed a new vaccine for preventing the flu. The company claims that less than 5% of adults who use its vaccine will get the flu. To test the claim, researchers give the vaccine to a random sample of 1000 adults. Of these, 43 get the flu.
Ho: p > 0.05
Ha: p < 0.05
What is a Type I Error in this setting?
What is a Type II Error in this setting?
Which would be a more serious mistake in this setting—a Type I error or a Type II error?
Deciding that less than 5% of adults who use the vaccine will get the flu and less than 5% actually get the flu.
Deciding that less than 5% of adults who use the vaccine will get the flu, so the vaccine is used and 5% or more will actually get the flu.
Deciding that 5% or more of adults who use the vaccine will get the flu, so the vaccine is not used and less than 5% will actually get the flu.
Deciding that 5% or more of adults who use the vaccine will get the flu so the vaccine is not used and 5% or more will actually get the flu.
Type I Error
Type II Error
A college president says, “More than two-thirds of the alumni support my firing of Coach Boggs.” The president’s statement is based on 200 emails he has received from alumni in the past three months.
The college’s athletic director wants to perform a test of
H0: p = 2/3 versus Ha: p > 2/3, remember: 2/3 = 0.667
where p = the true proportion of the college’s alumni who favor firing the coach.
Have the conditions been met for calculating a significance test?
Select all correct answers:
Zenon decided to investigate whether students at his school prefer name-brand potato chips to generic potato chips. He randomly selected 50 students and had each student try both types of chips, in random order.
Overall, 34 of the 50 students preferred the name-brand chips.
Zenon wants to perform a test at the α = 0.05 significance level of
H0: p = 0.5 versus Ha: p > 0.5,
where p = the true proportion of all students at his school who prefer name-brand chips.
Sample proportion =
Sampling distribution standard error =
Standardized test statistic (z statistic) =
What is the p value for having this sample proportion given the null value is true?
What is the conclusion based on the P value above?
Suppose that you want to perform a test at the α = 0.10 significance level of
H0 : μ = 5
Ha: μ ≠ 5
A random sample of size n = 20 from the population of interest yields mean= 5.21 and
Sx = 0.79.
Calculate the standard error for the sampling distribution:
Calculate the standardized test statistic:
Degrees of freedom =
Give the lower and upper probabilities for the P-value interval:
Based on an alpha level of 0.10, give the conclusion for the significance test.