This Question 1 pt This Test 10 pts Time Remaining 010435 0

This Question: 1 pt This Test: 10 pts Time Remaining: 01:04:35 0 of 10 complete A data set of 5 observations for Concession Sales per person (S) at a theater and Minutes before the movie begins results in the following estimated regression model. Complete parts a through c below Sales-4.4 +0.251 Minutes a) A 90% prediction interval for a concessions customer 10 minutes before the movie starts is ($5.81,$8.01). Explain how to interpret this interval. Choose the correct answer below A. 90% of all customers spend between $5.81 and $8.01 at the concession stand. B. 90% of the 5 observed customers 10 minutes before the movie starts can be expected to spend between $5.81 and $8.01 at the concession stand OC. There is a 90% chance that the mean amount spent by customers at the concession stand 10 minutes before the movie starts is between $5.81 and $8.01 D. 90% of customers 10 minutes before the movie starts can be expected to spend between $5.81 and $8.01 at the concession stand.

Solution

A pre­dic­tion inter­val is an inter­val asso­ci­ated with a ran­dom vari­able yet to be observed, with a spec­i­fied prob­a­bil­ity of the ran­dom vari­able lying within the inter­val. For exam­ple, I might give an 80% inter­val for the fore­cast of GDP in 2014. The actual GDP in 2014 should lie within the inter­val with prob­a­bil­ity 0.8. Pre­dic­tion inter­vals can arise in Bayesian or fre­quen­tist statistics.

A con­fi­dence inter­val is an inter­val asso­ci­ated with a para­me­ter and is a fre­quen­tist con­cept. The para­me­ter is assumed to be non- random but unknown, and the con­fi­dence inter­val is com­puted from data. Because the data are ran­dom, the inter­val is ran­dom. A 95% con­fi­dence inter­val will con­tain the true para­me­ter with prob­a­bil­ity 0.95. That is, with a large num­ber of repeated sam­ples, 95% of the inter­vals would con­tain the true parameter.

To estimate the uncertainty in our estimate of the conditional mean E(Y|X = x), we can construct a confidence interval for the conditional mean. But, the uncertainty in our estimate of Y when X = x is greater than our uncertainty of E(Y|X = x). Thus, the confidence interval for the conditional mean underestimates the uncertainty in our use of as an estimate of a value of Y|(X = x). Instead, we need what is called a prediction interval, which takes into account the variability in the conditional distribution Y|(X = x) as well as the uncertainty in our estimate of the conditional mean E(Y|(X = x)).

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a)

Based on above theory, we can say that the option (D) is correct because the prediction interval tells us about the interval of the predicted value for a new observation.

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b)

Again, the option (D) is correct because the confidence interval of mean tells us about the interval in which we can expect the value of Y to lie given a particular value of X.

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c)

Option (C) is correct.