p.439 – 441 #27 – 30, 34, 39, 43 – ) (a) We would be surprised to find 32% orange candies in this case. Very few of the simulations with sample size 25 had 32% or more orange candies. However, we would NOT be surprised to find 20% orange candies. This is very near the center of the distribution. (b) We would be surprised to find 32% orange candies in either case since neither simulation had many samples with 32% or more orange candies. However, it is even rarer when the sample size is 50.

p.439 – 441 #27 – 30, 34, 39, 43 – ) (a) The mean of the sampling distribution is the same as the population proportion, so it is (b)The standard deviation of the sampling distribution is √((0.15)(0.85)/25) = In this case, the 10% condition is met because it is very likely true that there are more than 250 candies. (c)The sampling distribution is not approximately Normal because np = 25(0.15) = 3.75 is less than 10. (d)The sampling distribution would now be approximately Normal with mean 0.15 and standard deviation , since np = and n(1-p) = are both at least 10.

p.439 – 441 #27 – 30, 34, 39, 43 – ) The 10% condition is not met here. The sample of 50 is more than 10% of the population (which is size 316) 44)C 46.) B

Section 7.2 was about sample _________________________. We use proportions most often when we are interested in categorical variables (ex – What proportion of US adults attend church?) However, when we have quantitative data (ex – hours of sleep per night, income of a household, blood sugar level, etc.) we are more interested in other statistics, such as the ________________________, ________________________, or ________________________ __________________________. The sample means are just averages of observations. This lesson describes the sampling distribution of the mean of the responses in an SRS. proportions IQR mean Standard deviation median

Averages (means) are _______________ variable than individual observations. Averages (means) are _______________ Normal than individual observations. less more

less four ten

Moviegoing Students

Activity Search for “Rice University sampling distributions applet.” Listen for the prompts, discover the patterns, and be ready to discuss.

If an SRS of size n from a population has the Normal distribution with mean  and standard deviation σ, then the sample mean ________ has the Normal distribution N ( ________, ________ ), provided the 10% condition is satisfied!

Buy Me Some Peanuts At the P. Nutty Peanut Company, dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of weights in the jars is approximately Normal, with a mean of 16.1 oz and a standard deviation of 0.15 oz. a)Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding its contents weigh less than 16 oz or randomly selecting 10 jars and finding that the average contents weigh less than 16 oz. b)Find the probability of each event described above.

Solution—part (a) Since averages are less variable than individual measurements, you would expect the sample mean of 10 jars to be closer, on avg, to the true mean of 16.1 oz. Thus, it is more likely that a single jar would weigh less than 16 oz than for the AVERAGE of 10 jars to be less than 16 oz!

Solution—part (b) b) Let X = weight of contents of randomly selected jar of peanuts. X is Normal with N(16.1, 0.15 ). Find P(X < 16). Standardize, draw curve and shade! z = ; P(z < ) = normalcdf( -E99, 16, 16.1, 0.15) =

Solution – part (b)

Draw an SRS of size n from ANY population with mean  and FINITE standard deviation σ. (It does not matter what shape the population distribution has!) When n is large, the sampling distribution of the sample mean __________ is close to the Normal distribution with mean _________ and standard deviation __________.

Although many populations have roughly Normal distributions, few are exactly Normal. This theorem is true no matter what shape the population distribution has, as long as the population has a finite standard deviation. How large is large enough? It depends on the population distribution … more observations are required if the shape of the population distribution is far from Normal! In most cases____________. n ≥ 30

EXAMPLE: Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows the distribution whose density curve is shown in the margin. This distribution is strongly right-skewed, and the most likely outcomes are closest to 0. The mean time is μ = 1 hour and the standard deviation is σ = 1 hour. In the coming week, your company will service an SRS of 70 air-conditioning units in the city. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough? (Follow the four-step process.)

EXAMPLE: Suppose that the number of texts sent during a typical day by a randomly selected high school student follows a right-skewed distribution with a mean of 15 and a standard deviation of 35. Assuming the students at your school are typical texters, how likely is it that a random sample of 50 students will have sent more than a total of 1000 texts in the last 24 hours? Use the four-step process.

Read Textbook p. 442 – 453 Do exercises p. 454 – 457 #49, 51, 54, 55, 59, 65 – 68 Check answers to odd problems