Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 13 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.36 gram. When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.) zc = (a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) σ is known σ is unknown uniform distribution of weights normal distribution of weights n is large (c) Interpret your results in the context of this problem. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20. The probability to the true average weight of Allen's hummingbirds is equal to the sample mean. There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region. There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region. (d) Which equation is used to find the sample size n for estimating μ when σ is known? n = zσ E σ n = zσ σ E 2 n = zσ E σ 2 n = zσ σ E Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.10 for the mean weights of the hummingbirds. (Round up to the nearest whole number.) hummingbirds

The next two questions (7 and 8) refer to the following:

The weight of bags of organic fertilizer is normally distributed with a mean of 60 pounds and a standard deviation of 2.5 pounds.

7. What is the probability that a random sample of 33 bags of organic fertilizer has a total weight between 1963.5 and 1996.5 pounds?

8. If we take a random sample of 9 bags of organic fertilizer, there is a 75% chance that their mean weight will be less than what value? Keep 4 decimal places in intermediate calculations and report your final answer to 4 decimal places.

The next two questions (8 and 9) refer to the following:

Question 10 and 11

Suppose that 40% of students at a university drive to campus.

10. If we randomly select 100 students from this university, what is the approximate probability that less than 35% of them drive to campus?

Keep 6 decimal places in intermediate calculations and report your final answer to 4 decimal places.

11. If we randomly select 100 students from this university, what is the approximate probability that more than 50 of them drive to campus?

Keep 6 decimal places in intermediate calculations and report your final answer to 4 decimal places.

12. Suppose that IQs of adult Canadians follow a normal distribution with standard deviation 15. A random sample of 30 adult Canadians has a mean IQ of 112.

We would like to construct a 97% confidence interval for the true mean IQ of all adult Canadians. What is the critical value z* to be used in the interval? (You do not need to calculate the calculate the confidence interval. Simply find z*. Input a positive number since we always use the positive z* value when calculating confidence intervals.)

Report your answer to 2 decimal places.

We made the shift from working with discrete random variables back in Unit 2 to now working with continuous random variables. Highly important for the rest of the course is our ability to use the Normal Distribution, in which we either find a probability from a range of values of the normally distributed variable X or we find a range of values from a given probability.

We will use shorthand notation (review LM 09 for how to use this notation) and probability notation for random variables (review LM 09 and LM 07 as needed) when working with normally distributed random variables. Suppose the vitamin C content of a particular variety of orange is distributed normally with mean 720 IU and standard deviation 46 IU. If we designate

X = the vitamin C content of a randomly selected orange,

then our shorthand notation is

X~N(720 IU, 46 IU).

Use this distribution of vitamin C content to answer the following questions:

3pt 1) What is the probability that a randomly selected orange will have less than 660 IU? Using X as the random variable, state your answer as a probability statement using the probability notation developed in the learning module.

3pt   2) What is the 80th percentile of the of the distribution of vitamin C content of the oranges?

1pt   3) What proportion of oranges exceed the vitamin C content you found in part (2) above?

3pt   4) What range of vitamin C content values represent the middle 80% of the distribution? State your answer as a probability statement using the probability notation developed in the learning module.

Extra Credit:

3pt EC   Suppose Y~N( 280 mg, 20 mg). Find Y₁ such that P( Y > Y₁) = 0.0250. State your answer in the form of a complete sentence without using any probability notation.

EXCEL CAN BE USED

You still work at that Starbucks. Due to COVID-19, the business is slow. As the manager, you had to ask two employees of yours to stay home and wait for more shifts to open. Meanwhile, you are bored. So you look into historical data from the store and dig out the following:

Customers spent an average of $4.18 on iced coffee with a standard deviation of $0.84.

43% of iced-coffee customers were women.

21% were teenage girls.

In order to increase sales, Starbucks start to offer a half-priced Frappuccino beverage between 3 pm and 5 pm for a limited time. One month after the marketing period ends, you survey 50 of your iced-coffee customers and find that:

They spent an average of $4.26 on the drink.  

46% were women.

34% were teenage girls.

Since these numbers are different from what the store historical data has told, you wonder whether the store data are outdated.  

1. What's the probability that customers spend an average of $4.26 or more on iced coffee? Round your answer to four decimal places. Based on this probability, are you convinced that the store data on the average spend on the drink are outdated? Hint: If the probability is < 0.05, the store data are outdated.  

2. What's the probability that 46% or more of iced-coffee customers are women? Round your answer to four decimal places. Based on this probability, are you convinced that the store data on women are outdated? Hint: If the probability is < 0.05, the store data are outdated.  

3. What's the probability that 34% or more of iced-coffee customers are teenage girls? Round your answer to eight decimal places. Based on this probability, are you convinced that the store data on teenage girls are outdated? Hint: If the probability is < 0.05, the store data are outdated.  

1. Archway Adventures has compiled data for 2017 on customer orders by region and product group as follows:

Let us assume that these are representative of the pattern of orders that they anticipate seeing in 2018.

1. What is the probability that a randomly selected order will be from North America?
2. What is the probability that an order is from North America and it is for a game?
3. What is the probability that an order is from outside North America and Europe or is for something other than a toy or game?
4. Among orders from the “rest of the World”, what is the probability it is for a toy?
5. Among orders for toys, what is the probability that the order comes from the “Rest of the World”?
6. Are the events “Toy” and “Rest of the World” independent? Explain.
7. The rows of the table are mutually exclusive categories. How would our summary be complicated if some orders were for multiple products?

2. 54.5% of Archway’s orders come from North America, 31.5% from Europe and the remaining 14% from the rest of the world. Among North American customers, 3% complained about late or lost deliveries. Similarly, complaints were received from 8% of European customers and 15% of customers from the rest of the world.

1. Construct a probability tree that shows all combinations of where orders originated and whether customers complained. Label all branches in terms of what the event is on the branch and the corresponding probability (e.g., P(A), PA’), P(B|A), P(B’|A),…). Also determine the probability for each ending joint event (e.g., P(Europe and Complain)).
2. Summarize all of the joint probabilities in a probability table.
3. What percentage of Archway customers complained about delivery problems?
4. Among those who complained, what percentage were from Europe?

According to a survey in a​ country, 18​% of adults do not own a credit card. Suppose a simple random sample of 900 adults is obtained. ​(b) What is the probability that in a random sample of 900 ​adults, more than 22​% do not own a credit​ card? The probability is what.? ​(Round to four decimal places as​ needed.) Interpret this probability. If 100 different random samples of 900 adults were​ obtained, one would expect nothing to result in more than 22​% not owning a credit card. ​(Round to the nearest integer as​ needed.).  

(c) What is the probability that in a random sample of 900 ​adults, between 17​% and 22​% do not own a credit​ card?

Interpret this probability. If 100 different random samples of 900 adults were​ obtained, one would expect ? to result in between 17​% and 22​% not owning a credit card.

​(Round to the nearest integer as​ needed.)

Would it be unusual for a random sample of 900 adults to result in 153 or fewer who do not own a credit​ card? Why? Select the correct choice below and fill in the answer box to complete your choice

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D.The result

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nothing​,

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30% of what number is 95?

Prove that there exists a negative number.