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PS 5.1, 5.2, 5.3 1) The density histogram to the right gives the relative frequency distribution for the amount of sodium in the fries from the fast food restaurants, with the observations for sodium content grouped into classes. a) What is the class width? Since there are 5 rectangles above the x-axis between 500 and 1000, the class widths must be 100. b) What is the approximate relative frequency of the observation that the fries had anywhere between 1400 and 1500mg of sodium? If we randomly select a restaurant from the sample, what is the probability that a serving of its fries has … c) … between 1400 and 1500mg of sodium? d) … more than 300mg of sodium? e) … between 500 and 700mg of sodium? f) … between 100 and 200mg of sodium or between 1500 and 1600mg of sodium? 2) A deck of cards has 52 different cards. Assume that when randomly selecting one of these cards, each is equally likely to be selected. There are 4 suits of cards - each containing 13 cards - and there are two colors: hearts and diamonds are red, clubs and spades are black. Each suit has 3 face cards: jack, queen, and king. If you randomly select a card from the deck… a) … what is the probability that the card is a diamond card or a face card? b) … what is the probability that the card king, given that it is a face card? c) … what is the probability that the card is an Ace, given that it is not a face card? 3) Suppose that for a simple random sample of 100 pole-vaulters, the data on their heights form a unimodal, approximately symmetric distribution with a mean of 5 feet, ten inches, and a standard deviation of one inch. If we randomly select one of these pole-vaulters, what is the probability that his height will be between 5 feet, 9 inches, and 5 feet, 11 inches? 4) Suppose an experiment consists of rolling a single die 3 times. a) Is the result of any one roll affected by the result of the previous roll(s)? b) What is the probability of the outcome where we roll a 1, a 4, and then a 5? c) Which outcome is more likely? Rolling 6 three times in a row or rolling a 2, then a 5, and then a 3? 5) A simple random sample of 720 students from nearby 4-year institutions was gathered to conduct as part of a study on aging and education. The data is illustrated in the two-way table below: I changed the layout for this table for use in future classes, but it contains the same info. If we randomly select one of the students from the sample, what is the probability that the student is… a) …a junior? b) …between 22 to 27 years old? c) …a junior between 22 to 27 years old? d) …a junior or between 22 to 27 years old? e) …a junior, given that the student is between 22 to 27 years old? f) …between 22 to 27 years old, given that the student is a junior? TM 7.1 6) The local restaurant is having a dinner promotion: for $17.99 you get a drink, a salad, a cup of soup, an entrée, and a dessert. If there are 5 different drinks, 2 salads, 4 types of soup, 9 different entrées, and 3 desserts to choose from, how many different ways can you order a dinner special? TM 7.2 7) Evaluate the following expressions without the factorial button on your calculator: a) 5! b) 1792! 1790! 8) Evaluate the expression: 17?14 9) A local television talent show has enough time to showcase 6 different performers. If 32 performers try out, how many different ways can the 1st, 2nd, 3rd, 4th, 5th, and 6th time slots be filled? TM 7.3 10) Evaluate the expression: 21?16 11) A lottery game involves randomly selecting 6 balls from a large drum containing 64 balls numbered from 1 to 64. All the balls are equally likely to be selected. For each ticket you purchase, you get to choose one combination of 6 different numbers. If any of your 6 number combinations match all the numbers on the 6 balls selected, you win – and the order in which the balls were selected does not matter. a) How many different 6 number combinations can be drawn from the drum? b) Suppose you purchase 10,000 tickets. Express your odds of winning in both fraction form and decimal form, rounded to three decimal places. SS – General Discrete Probability Models & SS – Expected Value and Standard Deviation 12) 48 students in a first year marketing course were asked how many social media platforms they had profiles for. 4 students said they had 1 profile, 24 students said they had 2 profiles, 12 students said they had 3 profiles, and 8 students said they had 4 profiles. Answer the following questions to produce a discrete probability model that illustrates the experiment of randomly selecting one of these students and determining how many social media profiles the student has. a) Is the collection of observations a continuous or discrete data set? b) List the possible outcomes: ? = c) Define a discrete random variable for this situation. d) Complete the following table and use it to answer the questions below PDF ? ?(? = ?) ? ∙ ?(? = ?) (? − ?)? (? − ?)??(? = ?) 1 2 3 4 e) Find the following probabilities: i) ?(? ≥ 2) ii) ?(1 ≤ ? ≤ 3) iii) ?(? = 1 OR 3) g) What is the expected value of the discrete random variable? i) What is the standard deviation of the discrete random variable? PS 4.1 1) a) Find the mean and median of the set {−2, 5, 3, 7, −1, −1, 2, 4, 7}. b) Build a histogram for the variable “On-time Percentage” in the data set “Flight Delay Data For July 2014.” Just based on the shape of the distribution alone, would you say that the mean is an acceptable measure of center in this case? Why or why not? 2) Build a histogram for the variable “Horsepower” in the data set “Lab 1b Data” a) Which is a more appropriate measure of center here, the mean or the median? b) Are there any outliers? What are they? (Hint: Build a boxplot for the same variable). c) Which measure of center did these outlier affect more, the mean or the median? PS 4.2 3) Find the standard deviation of the data set {−1, 1, 3, 3, 4} by hand. (no calculator, no StatCrunch) 4) For a unimodal and approximately symmetric distribution… a) …which of the two is likely to be a more useful measure of center, the mean or median? Explain. b) …which of the two is more likely a more useful measure of spread, the range or standard deviation? Explain. c) …Does the empirical rule apply? d) What percent of the observations are supposed to fall within 1 standard deviation of the mean? e) What percent of the observations are supposed to fall within 2 standard deviations of the mean? 5) Suppose you have a unimodal, approximately symmetric distribution with 200 observations. Suppose its mean is 235, its standard deviation is 4, and that no other information is known about this distribution. Answer the following questions by giving the best estimates. a) About how many observations fall in the interval (231, 239)? b) How many should fall in the interval (227, 243)? c) How many unusual observations will there likely be? PS 4.3 6) The two boxplots represent the home value distributions for two California cities. Use them to answer the questions below. a) For each city, estimate ?1, the median, ?3, and the interquartile range (IQR) b) Based on your answer for part (a), in which of the two cities would you the housing is more affordable? c) Which of the two cities has the three most expensive homes? d) How could your answers to parts (b) and (c) both be true? (hint: think about the relationship between the median and outliers) 7) Use the coordinate system below to draw a boxplot – by hand – for the distribution whose observations are the numbers 4, 5, 6, 6, 7, 9, 10, 10, 12, 17. To do this, you will need to consider the quartiles, median, IQR, fences, etc. (for more help, see example 4 on p266 of the Pathways.. text). TM 5.1, 5.2 8) Let ? = {?, ?, ?, ?, ?} a) Find a subset of ?. b) Is ? = {?, ?, ?} a subset of ?? Is it a proper subset of ?? c) Is ? = {?, ?, ?, ?} a subset of ?? Is it a proper