Vocabulary
States that the distribution of sample means will be approximately normal if the sample size is large enough, n is greater than or equal to 30, whether or not the population distribution is normal.
What is the Central Limit Theorem
Suppose a large local bakery sells cupcakes, and historically, 20% of the cupcakes sold are chocolate flavored. Imagine taking an SRS, from all the cupcakes they make in a day, of 50 cupcakes from the bakery. What is the mean of the sampling distribution of p-hat and why?
The mean of p-hat = 0.20 because the sample was randomly selected, the center of the sampling distribution will be equal to the population proportion.
Scott's movie collection includes hundreds of short films. The distribution of the run times for these films has a mean of 225 seconds and a standard deviation of 60 seconds. Suppose we randomly select a simple random sample (SRS) of 10 films from this collection and calculate the mean run time x-bar of these films. Calculate the mean of the sampling distribution.
the mean of the sampling distribution is 225.
In a bustling metropolitan area, the distribution of apartment rental prices in Uptown has a mean of $2,300 per month and a standard deviation of $600. Meanwhile, in Downtown, the distribution of rental prices has a mean of $2,000 per month and a standard deviation of $550. Independent random samples of 10 apartments from each area are selected. Let X1 and X2 represent the sample mean rental prices for Uptown and Downtown, respectively. Calculate the mean and the standard deviation of the sampling distribution X1 - X2
Mean: $300
SD: $257.39
The display of several sample proportions (p-hat) from random samples of size n, each taken from the same population, where the mean of all sample proportions (p-hat) is the true population proportion.
Sampling Distribution of Sample Proportions
Suppose a large local bakery sells cupcakes, and historically, 20% of the cupcakes sold are chocolate flavored. Imagine taking an SRS, from all the cupcakes they make in a day, of 50 cupcakes from the bakery. Find the standard deviation of the sampling distribution of p-hat. Check the 10% condition and explain how it is a reasonable assumption.
SD = 0.0566
.5 <= 0.10(all cupcakes in the bakery)
This inequality is a reasonable assumption because we would assume that a large local bakery would make over 500 cupcakes in a day.
Scott's movie collection includes hundreds of short films. The distribution of the run times for these films has a mean of 225 seconds and a standard deviation of 60 seconds. Suppose we randomly select a simple random sample (SRS) of 10 films from this collection and calculate the mean run time x-bar of these films. Calculate the standard deviation of the sampling distribution.
Standard Deviation is 18.97
A researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic math test. Suppose we give a basic math test to a random sample of 60 high school graduates and an independent random sample of 75 high school dropouts. Let Pg and Pd be the sample proportions of graduates and dropouts, respectively, who pass the test. Calculate the mean and the standard deviation of Pg-Pd
P-hat = 0.4
SD = 0.077
The display of several sample means (x-bar) from random samples size n, each taken from the same population, where the mean of all sample means (x-bar) is the true population mean.
Sampleing distribution of Sample Means
Suppose a large local bakery sells cupcakes, and historically, 20% of the cupcakes sold are chocolate flavored. Imagine taking an SRS, from all the cupcakes they make in a day, of 50 cupcakes from the bakery. Is the sampling distribution of p-hat approximately normal?
Checking the large counts condition:
50(.2)>=10 and 50(.8)>=10
The condition is satisfied, therefore we can say tha the sampling distribution of p-hat will be approximately normal.
A particular restaurant (that shall remain nameless) is suspected of undercooking its burgers. The restaurant claims that its burgers are cooked to a medium doneness, with an average internal temperature of 160 degrees Fahrenheit and a standard deviation of 5 degrees Fahrenheit. Assume that the restaurant's claim is accurate and that the distribution of internal temperatures follows a normal distribution. What is the probability that a single randomly selected burger is cooked to an internal temperature of 158 degrees or less?
P(X <= 158) = .3446
In a clinical trial investigating the effectiveness of a new medication for allergies in dogs, 60 dogs were randomly assigned to two groups: 30 dogs went to Group M, where they received the new medication, while 30 dogs went to Group P, where they received a placebo. After a month of treatment, the severity of allergy symptoms in each dog was assessed using a standardized scale. The mean score for Group M was 3.5 units with a standard deviation of 1.2 units, while the mean score for Group P was 4.2 units with a standard deviation of 1.5 units. A higher score means that more allergy symptoms were observed. What is the probability that the difference in the mean scores between the two groups (Group M − Group P) is greater than 0? Assume all conditions hold.
States that, in the long run, as we repeat a random process over and over, the proportion of times that an event occurs gets closer and closer to 1.
Law of Large Numbers
The National Health Survey (NHS) reports that 80% of adults in a certain region engage in regular physical exercise. To determine if this trend holds true in a specific neighborhood, a local health organization conducts a survey of a simple random sample of 150 adults, in a neighborhood with over 2000 adults. Among these adults, 115 report engaging in regular physical exercise. Assume all 3 conditions hold, what is the probability that we get this result or lower, if we assume tha tthe NHS's result holds true in our neighborhood?
P(p-hat <0.77) = 0.1796
A particular restaurant (that shall remain nameless) is suspected of undercooking its burgers. The restaurant claims that its burgers are cooked to a medium doneness, with an average internal temperature of 160 degrees Fahrenheit and a standard deviation of 5 degrees Fahrenheit. Assume that the restaurant's claim is accurate and that the distribution of internal temperatures follows a normal distribution.
A secret shopper comes in at random times throughout the day to test the internal temperature of the burgers. They select an SRS of 25 burgers and calculate the sample mean. What are the mean and standard deviation of the resulting sampling distribution?
Mx-bar = 160
SD: 5/sqrt(5) = 1
A study is conducted to compare the effectiveness of two study techniques, A and B, in improving test scores. For Technique A, a random sample of 80 students yields an average score of 85 with a standard deviation of 10. For Technique B, a random sample of 100 students yields an average score of 82 with a standard deviation of 8. Calculate the probability that the difference in the average scores between Technique A minus Technique B is greater than 4. Assume all conditions are met.
The probability that the difference in the average scores between technique A and technique B is greater than 4 is 0.234
The cases we examine in order to understand something about a larger population; a subset of the population.
A fast-food chain claims that 85% of its drive-thru orders are completed within 3 minutes. To verify this claim, a manager selects a random sample of 150 drive-thru orders from the past week for analysis. Among these, 123 orders were completed within 3 minutes. If the company's claim of completing 85% of orders within 3 minutes is accurate, what is the probability that the proportion in an SRS of 150 orders is 0.82 or less? Assume all 3 conditions hold to proceed.
A particular restaurant (that shall remain nameless) is suspected of undercooking its burgers. The restaurant claims that its burgers are cooked to a medium doneness, with an average internal temperature of 160 degrees Fahrenheit and a standard deviation of 5 degrees Fahrenheit. Assume that the restaurant's claim is accurate and that the distribution of internal temperatures follows a normal distribution.
The secret shopper in part (b) obtains a sample mean of x̅= 158 degrees Fahrenheit. What is the probability that a random sample of 25 burgers produces a sample mean amount of 158 degrees or less? (Assume all conditions are met)
0.023
Dane fails to study for his statistics final. The final has 100 multiple-choice questions, each with 5 choices. Assume all questions are independent and that Dane has given himself over to the Gods of Chance and randomly guesses on each of the 100 questions. What is the probability that Dane will get between a 25% and 50% on the test?
P(0.25 < p < .5) = 0.1056