BUSA 251 - BUS Stats
A small sample of computer operators shows monthly incomes of $1,950, $1,775, $2,060, $1,840, $1,795, $1,890, $1,925, and $1,810. What are these ungrouped numbers called?
Raw data
Descriptive statistics are used to find out something about a population based on a sample. When statisticians analyze sample data in order to draw conclusions about the characteristics of a population, this is referred to as
Stat inference
When dividing a population into subgroups so that a random sample from each subgroup can be collected, what type of sampling is used?
Stratified random sampling
When testing for differences between treatment means, the degrees of freedom for the t-statistic are ________.
(n − k)
The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a population standard deviation of 6 hours. Suppose we select a random sample of 144 current students. What is the standard error of the mean?
.50 To find the standard error of the mean, use the formula: σX¯¯¯=σ/n−−√ =6/144 −−−−√=0.5
The Intelligence Quotient (IQ) test scores for adults are normally distributed with a population mean of 100 and a population standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample exceeds 104?
0.0294 WORD DOC We use the formula z=X¯¯¯−μσ/n√=104−10015/50√=1.89z=X¯-μσ/n=104-10015/50=1.89 . Then, using the "areas under the normal curve" table, P(z ≥ 1.89) = 0.5000 − 0.4706 = 0.0294.
David's gasoline station offers 4 cents off per gallon if the customer pays in cash and does not use a credit card. Past evidence indicates that 40% of all customers pay in cash. During a one-hour period, 15 customers buy gasoline at this station. What is the probability that at least 10 pay in cash?
0.033
When doing research, knowing the population mean and other population parameters is essential.
F
For the following distribution. xP(x)0 0.130 1 0.346 2 0.346 3 0.154 4 0.026 What is the mean of the distribution?
1.604 To find the mean: µ = 0(0.13) + 1(0.346) + 2(0.346) + 3(0.154) + 4(0.026) = 1.604
The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the distribution's mean?
135 minutes The mean of a uniform distribution is the center of the interval between the minimum and maximum values. In this case, it is 135 minutes, found by (120 + 150)/2.
Refer to the following distribution of commissions. Monthly CommissionsClass Frequencies$600 up to $8003800 up to 1,00071,000 up to 1,200111,200 up to 1,400121,400 up to 1,600401,600 up to 1,800241,800 up to 2,00092,000 up to 2,2004 For the preceding distribution, what is the midpoint of the class with the greatest frequency?
1500
A sample of small bottles and their contents has the following weights (in grams): 4, 2, 5, 4, 5, 2, and 6. What is the sample variance of bottle weight?
2.33
Refer to the following frequency distribution of days absent during a calendar year by employees of a manufacturing company. How many employees were absent for 6 up to 12 days?
20
Refer to the following distribution of commissions. Monthly CommissionsClass Frequencies$600 up to $8003800 up to 1,00071,000 up to 1,200111,200 up to 1,400121,400 up to 1,600401,600 up to 1,800241,800 up to 2,00092,000 up to 2,2004 What is the class interval?
200
Refer to the following frequency distribution of days absent during a calendar year by employees of a manufacturing company. Days AbsentNumber of Employees0 up to 3603 up to 6316 up to 9149 up to 12612 up to 152 How many employees were absent six or more days?
22
Refer to the following frequency distribution of days absent during a calendar year by employees of a manufacturing company. Days AbsentNumber of Employees 0 up to 3 13 3 up to 6 23 6 up to 9 29 9 up to 12 30 12 up to 15 12 How many employees were absent fewer than six days?
36
Production of passenger cars in Japan increased from 3.74 million in 1999 to 6.64 million in 2009. What is the geometric mean annual percent increase?
5.9
For the following distribution of heights, what are the limits for the class with the greatest frequency? Heights60" up to 65"65" up to 70"70" up to 75"Frequency107020
65 and up to 70
The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. What is the probability of a weight between 415 pounds and the mean of 400 pounds?
Begin by finding the z-value corresponding to 415 pounds, from formula z = (x − μ)/σ: z =(x − μ)/σ = (415 − 400)/10 = 1.5. Then, find the probability z is between 0 and 1.5 is 0.4332 by using the "areas under the normal curve" table.
The average score of 100 students taking a statistics final was 70 with a standard deviation of 7. Assuming a normal distribution, what test score separates the top 25% of the students from the lower 75% of students?
74.69
On a finance exam, 15 accounting majors had an average grade of 90. On the same exam, 7 marketing majors averaged 85, and 10 finance majors averaged 93. What is the weighted mean for all 32 students taking the exam?
89.84
Refer to the following breakdown of responses to a survey of "Are you concerned about being tracked while connected to the Internet?" ResponseFrequencyVery concerned140Somewhat concerned40No concern20 What type of chart should be used to describe the frequency table?
A bar chart
For each of the following indicate whether the random variable is discrete or continuous. a. The length of time to get a haircut. b. The number of cars a jogger passes each morning while running. c. The number of hits for a team in a high school girls' softball game. d. The number of patients treated at the South Strand Medical Center between 6 and 10 P.M. each night. e. The distance your car traveled on the last fill-up. f. The number of customers at the Oak Street Wendy's who used the drive-through facility. g. The distance between Gainesville, Florida, and all Florida cities with a population of at least 50,000.
A discrete probability distribution can take on only values that are clearly separated from each other. On the other hand, a continuous probability distribution can take on any value in a range. a.Continuous, time can be any of an infinitely large number of values. b.Discrete, the number of cars can only be whole numbers or clearly separated amounts. c.Discrete, the number of hits can only be whole numbers or clearly separated amounts. d.Discrete, the number of patients treated can only be whole numbers or clearly separated amounts. e.Continuous, distance can be any of an infinitely large number of values. f.Discrete, the number of customers can only be whole numbers or clearly separated amounts. g.Continuous, distance can be any of an infinitely large number of values.
Refer to the following breakdown of responses to a survey of room service in a hotel. ResponseFrequency Not satisfied20 Satisfied40 Highly satisfied60 What type of chart should be used to show relative class frequencies?
A pie chart
All possible samples of size n are selected from a population, and the mean of each sample is determined. What is the mean of the sample means?
It is the population mean.
A population consists of all the weights of all defensive tackles on a university's football team. They are Johnson, 204 pounds; Patrick, 215 pounds; Junior, 207 pounds; Kendron, 212 pounds; Nicko, 214 pounds; and Cochran, 208 pounds. What is the population standard deviation (in pounds)?
About 4
The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed. What percent of the students scored below 320?
Begin by finding the z-value corresponding to a score of 320: z = (320 − 500)/75 = −2.40. Then, using the "areas under the normal curve" table, find the probability z is less than −2.40 is 0.0082, by computing 0.5000 − 0.4918. Thus, 0.82% of the students scored below 320.
What is the area under the normal curve between z = −1.0 and z = −2.0?
Because the normal distribution is symmetric, the area between 0 and a negative z-value is the same as that between 0 and the corresponding positive z-value. Using the "areas under the normal curve" table, the area under the normal curve between the mean and z = −1.0 is 0.3413, and between the mean and z = −2.0 is 0.4772. By subtraction, the area between z = −2.0 and z = −1.0 is 0.4772 − 0.3413 = 0.1359.
The z-scores for X values greater than the mean are negative.
F
There are four levels of measurement: qualitative, quantitative, discrete, and continuous.
F
What type of variable is the number of auto accidents reported in a given month?
Discrete
A frequency table for qualitative data has class limits.
F
Categorizing voters as Democrats, Republicans, and Independents is an example of interval level measurement.
F
Descriptive statistics are used to find out something about a population based on a sample.
F
If a confidence interval for the difference between a pair of treatment means includes 0, then we reject the null hypothesis that there is no difference in the pair of treatment means.
F
If we select 100 persons from 25,000 registered voters and question them about candidates and issues, the 100 persons are referred to as the population.
F
Quartiles divide a distribution into 10 equal parts.
F
The 50th percentile of a distribution is the same as the distribution mean.
F
The central limit theorem states that for a sufficiently large sample, the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed with the mean of the sampling distribution equal to σ2 and the variance equal to σ2/n.
F
The order in which runners finish in a race would be an example of continuous data.
F
The standard error of the mean is also called the sampling error.
F
According to the PMQ Pizza Magazine (https://www.pizzatoday.com/pizzeria-rankings/2018-top-100-pizza-companies/), an estimate of pizza sales in the United States for the top 100 pizza companies was $44.1 billion in 2018. Below are the top 5 companies with the number of franchise units and total gross sales in $ millions. Domino's 14,346 12,252 Pizza Hut 16,173 12,034 Little Caesars Pizza 5,311 4,000 Papa John's International 5,021 3,695 California Pizza Kitchen 258 840 To complete this exercise, please access the data set with the top 100 pizza companies. a. Using the data set, compute the sales per unit. (Enter the answers in $ millions. Round your answers to 3 decimal places.) b. Construct a frequency distribution of companies based on total sales. (Enter the answers in $ millions.) c. Construct a frequency distribution of companies based on per unit sales. (Enter the answers in $ millions.)
For every company, divide Sales by Units to get the average annual sales per unit for each company. b.Since 26 = 64 < 100 < 128 = 27, 7 classes are recommended. The interval should be at least (12,252 − 21)/7 = 1,747.3 use 1,750 as a convenient value. Set the lower limit of the first class to be zero. c.Since 26 = 64 < 100 < 128 = 27, 7 classes are recommended. The interval should be at least (3.750 − 0.038)/7 = 0.530 use 0.5 ($500,000) as a convenient value. Set the lower limit of the first class to be zero.
Refer to the following breakdown of responses to a survey of room service in a hotel. ResponseFrequency Not satisfied20 Satisfied40 Highly satisfied60 What is the class with the greatest frequency?
Highly satisfied
Which statement is correct about the F-distribution?
It cannot be negative.
Your height and weight are examples of which level of measurement?
Ratio
A group of women tried five brands of fingernail polish and ranked them according to preference. What level of measurement is this?
Ordinal
The names of the positions in a corporation, such as chief operating officer or controller, are examples of what type of variable?
Qualitative
For a standard normal distribution, what is the probability that z is greater than 1.75?
Recall half the area, or probability, is above the mean, and half is below. Using the standard normal probability distribution table, the area under the normal curve between 0 and 1.75 is 0.4599. Therefore, the probability that z is greater than 1.75 is 0.0401, found by 0.5000 − 0.4599.
The average score of 100 students taking a statistics final was 70, with a standard deviation of 7. Assuming a normal distribution, what test score separates the top 5% of the students from the lower 95% of the students?
Recall the area under the normal curve to the right of the mean is 0.5000. The area between the mean and the dividing point is 0.4500, found by 0.5000 − 0.0500. Now refer to the "areas under the normal curve" table. Search the body of the table for the area closest to 0.4500. The closest area is 0.4505. Move to the margins from this value and read the z-value of 1.65. Finally, the dividing point is 81.55, found by: score of top 5% = x + zσ = 70 + 1.65(7).
Which one of the following is not a condition of the binomial distribution?
Sampling at least 10 trials
What is the difference between a sample mean and the population mean called?
Sampling error
Suppose we select every fifth invoice in a file. What type of sampling is this?
Systematic
A class interval can be determined by subtracting the lower limit of a class from the lower limit of the next higher class.
T
A group of normal distributions can have equal arithmetic means but different standard deviations.
T
A sample is a portion or part of the population of interest.
T
A test statistic is a value computed from sample information that is used to test the null hypothesis.
T
For an ANOVA test, rejecting the null hypothesis does not identify which treatment means differ significantly.
T
In a bar chart, the heights of the bars represent the frequencies in each class.
T
In a bar chart, the horizontal axis is usually labeled with the values of a qualitative variable.
T
Statistics is defined as a body of techniques used to facilitate the collection, organization, presentation, analysis, and interpretation of information for the purpose of making better decisions.
T
The CIA World Factbook cited these numbers for the United States: The birthrate is 13.66 births per 1,000 of the population. The average life expectancy for females is 81.17 years. Approximately 316.7 million persons reside in the United States. Each of these numbers is referred to as a statistic.
T
The F-distribution's curve is positively skewed.
T
The Greek letter used to represent the probability of a Type I error is alpha (α).
T
The level of significance is the probability of rejecting the null hypothesis when it is actually true.
T
The mean of a probability distribution is called its expected value.
T
The number of children in a family is a discrete variable.
T
The principal difference between the interval and ratio scale is that the ratio scale has a meaningful zero point.
T
The probability of a particular outcome must always be between 0.0 and 1.0 inclusive.
T
When referring to the normal probability distribution, there is not just one; there is a "family" of normal probability distributions.
T
Within plus and minus one standard deviation of the mean, the area under any normal curve is about 68%.
T
What statistics are needed to draw a box plot?
The minimum, maximum, median, first and third quartiles
What does the interquartile range describe?
The range of the middle 50% of the observations
The manager of a computer software company wishes to study the number of hours per week senior executives by type of industry spend at their desktop computers. The manager selected a sample of five executives from each of three industries. At the 0.05 significance level, can she conclude there is a difference in the mean number of hours spent per week by industry?
WORD DOC
For the following distribution. xP(x)00.13010.34620.34630.15440.026 What is the variance of the distribution?
To find the variance: σ2 = (0 − 1.604)2(0.13) + (1 − 1.604)2(0.346) + (2 − 1.604)2(0.346) + (3 − 1.604)2(0.154) + (4 − 1.604)2(0.026) = 0.9643.
A recent national survey found that high school students watched an average (mean) of 7.5 movies per month with a population standard deviation of 0.9. The distribution of number of movies watched per month follows the normal distribution. A random sample of 38 college students revealed that the mean number of movies watched last month was 7.0. At the 0.05 significance level, can we conclude that college students watch fewer movies a month than high school students?
WORD DOC
A sample of 31 observations is selected from a normal population. The sample mean is 11, and the population standard deviation is 3. Conduct the following test of hypothesis using the 0.05 significance level.
WORD DOC
According to the Census Bureau, 3.04 people reside in the typical American household. A sample of 19 households in Arizona retirement communities showed the mean number of residents per household was 2.97 residents. The standard deviation of this sample was 1.33 residents. At the 0.05 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.04 persons?
WORD DOC
Arbitron Media Research Inc. conducted a study of the iPod listening habits of men and women. One facet of the study involved the mean listening time. It was discovered that the mean listening time for a sample of 9 men was 43 minutes per day. The standard deviation was 21 minutes per day. The mean listening time for a sample of 8 women was also 43 minutes, but the standard deviation of the sample was 13 minutes. At the 0.10 significance level, can we conclude that there is a difference in the variation in the listening times for men and women?
WORD DOC
At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, "You can average $83 a day in tips." Assume the population of daily tips is normally distributed with a standard deviation of $4.07. Over the first 45 days she was employed at the restaurant, the mean daily amount of her tips was $84.86. At the 0.05 significance level, can Ms. Brigden conclude that her daily tips average more than $83?
WORD DOC
In New York State, the mean salary for high school teachers in 2017 was $100,610 with a standard deviation of $9,600. Only Alaska's mean salary was higher! Assume New York's state salaries follow a normal distribution.
WORD DOC
Information from the American Institute of Insurance indicates the mean amount of life insurance per household in the United States is $132,000. This distribution follows the normal distribution with a standard deviation of $40,000.
WORD DOC
Power +, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows the normal probability distribution with a mean of 36 hours and a standard deviation of 5.8 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 16 batteries.
WORD DOC
The City of Maumee comprises four districts. Chief of Police Andy North wants to determine whether there is a difference in the mean number of crimes committed among the four districts. He examined the records from six randomly selected days and recorded the number of crimes.
WORD DOC
The mean income per person in the United States is $44,500, and the distribution of incomes follows a normal distribution. A random sample of 16 residents of Wilmington, Delaware, had a mean of $52,500 with a standard deviation of $9,500. At the 0.050 level of significance, is that enough evidence to conclude that residents of Wilmington, Delaware, have more income than the national average?
WORD DOC
Waterbury Insurance Company wants to study the relationship between the amount of fire damage and the distance between the burning house and the nearest fire station. This information will be used in setting rates for insurance coverage. For a sample of 30 claims for the last year, the director of the actuarial department determined the distance from the fire station (x) and the amount of fire damage, in thousands of dollars (y). The MegaStat output is reported here:
WORD DOC
A sample of 36 observations is selected from a normal population. The sample mean is 49, and the population standard deviation is 5. Conduct the following test of hypothesis using the 0.05 significance level.
Word
The speed with which utility companies can resolve problems is very important. GTC, the Georgetown Telephone Company, reports it can resolve customer problems the same day they are reported in 80% of the cases. Suppose the 14 cases reported today are representative of all complaints.
a-1.For a binomial distribution the mean is found multiplying the sample size by the probability of "success." So, the mean is 11.20 = 14(0.80).a-2.The variance is found by 14(0.80) (1 − 0.80) = 2.2400. Then the standard deviation is the square root of the variance. σ=14(0.80)(0.20)−−−−−−−−−−−√=1.4967σ=14(0.80)(0.20)=1.4967 b.P(10) = 0.1720, found by 14!10!4!(0.80)10(0.20)414!10!4!(0.80)10(0.20)4 which is formula (6-3) with n = 14, π = 0.80 and x = 10.c.P(10 or 11) = P(10) + P(11) = 0.4221, found by 0.1720 + 0.2501d.P(x ≥ 10) = P(10) + P(11) + P(12) +.... + P(14) or 0.8702, found by 0.1720 + 0.2501 + 0.2501 + 0.1539 + 0.0440Using software method:P(x ≥ 10) = 1−BINOM.DIST(10,14,0.80,1).
A sample of 25 undergraduates reported the following dollar amounts of entertainment expenses last year: 717 681 735 725 719 721 737 698 718 759 681 699 696 716 719 722 733 720 690 691 768 709 736 761 687 a. Find the mean, median, and mode of this information. (Round the "Mean" to 2 decimal places.) b. What are the range and standard deviation? (Round the "Standard deviation" to 2 decimal places.) c. Use the Empirical Rule to establish an interval which includes about 95 percent of the observations. (Round your answers to 2 decimal places.)
a.The mean is $717.52, found by $17,938/25. The median is $719 and there are two modes $681 and $719. b.The range is $87, found by $768 − 681, and the standard deviation is $24.02, found by the square root of 13,846.24/24. c.From $669.48 up to $765.56, found by $717.52 ± 2($24.02).
The Apollo space program lasted from 1967 until 1972 and included 13 missions. The missions lasted from as little as 7 hours to as long as 301 hours. The duration of each flight is listed below. 9 195 241 301 216 260 7 244 192 147 10 295 142 b-1. Find the mean of the flight times. (Round your answer to 2 decimal places.) b-2. Find the median of the flight times. c-1. Find the range of the flight times. c-2. Find the standard deviation of the flight times. (Round your answer to 2 decimal places.)
b-1.The mean is 173.77, found by 2259/13. b-2.The median is 195. c-1.The range is 294, found by 301 − 7. c-2.The standard deviation is 101.47, found by the square root of 133846/13.
Data show that the weight of an offensive linesman may be any weight between 200 and 350 pounds. The distribution of weight is based on a ______________.
continuous random variable
An example of a qualitative variable is
color of ink in a pen.
In ANOVA analyses, when the null hypothesis is rejected, we can test for differences between treatment means by ________.
constructing confidence intervals
In order to convert class frequency to relative class frequency, we
divide the frequency of the class by the sample size.
A university wishes to conduct a student survey. In one of the questions students are asked to mark their gender as either male or female. Gender is an example of the
nominal scale.
When a class interval is expressed as 100 up to 200,
observations with values of 200 are excluded from the class.
The length of a bridge, measured in meters, is an example of
quantitative data.
A table summarizing a set of data showing the fraction of the total number of items in several classes is a
relative frequency table.
Analysis of variance is used to ________.
simultaneously compare several population means
The upper and lower limits of a uniform probability distribution are __________.
the maximum and minimum values of the random variable