Confidence intervals and prediction intervals from simple linear regression
The managers of an outdoor coffee stand in Coast City are examining the relationship between coffee sales and daily temperature. They have bivariate data detailing the stand 's coffee sales (denoted by [pic], in dollars) and the maximum temperature (denoted by [pic], in degrees Fahrenheit) for each of [pic] randomly selected days during the past year. The least-squares regression equation computed from their data is [pic].
Tommorrow 's forecast high is [pic] degrees Fahrenheit. The managers have used the regression equation to predict the stand 's coffee sales for tomorrow. They now are interested in both a prediction interval for tomorrow 's
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The next term in the prediction interval formula is the standard error of the estimate, [pic]. It can be computed from the mean square error (MSE), which is given to be [pic]:
[pic].
The last part of the prediction interval formula consists of the square root of the sum of [pic] and a fairly long expression. We do not need to compute the long expression, though, because we were given its value: [pic]. We have
With this information, we can compute the [pic] prediction interval for the coffee sales given a maximum temperature of [pic] degrees Fahrenheit:
[pic].
Upon simplification, this is the interval whose lower limit is approximately [pic] and whose upper limit is approximately [pic]
2. Because there 's more precision involved in estimating the mean of a distribution than in predicting a particular observation from that distribution, we would expect the confidence interval to be narrower than the prediction interval. We can verify this by comparing the formulas for computing the intervals (shown near the top). As noted previously, the only difference between the prediction interval formula and the confidence interval formula is that the prediction interval formula has a [pic] in the sum underneath the square root, while the confidence interval formula does not. This makes the margin of error (the term following the "[pic]") greater in the prediction interval formula than in the confidence interval formula, which means that the
To compute the 90% prediction interval for all trading days during the study period, the formula ( , ) can be used. Referring to the question equals 0.1 and equals 0.05.
(21) You took a sample of size 21 from a normal distribution with a known standard deviation, . In order to find a 90% confidence interval for the mean, You need to find.
The case involves the decision to locate a new store at one of two candidate sites. The decision will be based on estimates of sales potential, and for this purpose, you will need to develop a multiple regression model to predict sales. Specific case questions are given in the textbook, and the necessary data is in the file named pamsue.xls.
A business wants to estimate the true mean annual income of its customers. It randomly samples 220 of its customers. The mean annual income was $61,400 with a standard deviation of $2,200. Find a 95% confidence interval for the true mean annual income of the business’ customers.
We have data out of 250 stores. The data include demographics, economics, sales of the stores, compositions of those sales as well as sales behavior per households. There are 31 variables being consider for each store and those variables range from sales,
Is this estimate centered about the parameter of interest (the parameter of interest is the answer for the mean in question 2)?
The purpose of this case is to determine which key variables drive Crusty Pizza Restaurant’s monthly profit and then forecast what the monthly profit would be for potential stores. Based off of this information we will be able to make a recommendation to Crusty Dough Pizza Restaurant on which stores they should open and which they avoid. The group was provided 60 restaurants’ data that included monthly profit, student population, advertising expenditures, parking spots, population within 20 miles, pizza varieties, and competitors within 15 miles. For the potential stores we were given all of this
Marketing is a competitive field that companies outdo each other to make a profit. Café Campesino is a retail-based company that assists farmers to sell their products in a fair and profitable trade. The American coffee industry is that which is on growth with more than 64% of American drinks an average of a cup of coffee a day. The coffee industry just like other agricultural products is affected by a host of factors from climatic variations to fluctuation of prices. This paper seeks to look at Café Campesino's marketing plan in the coffee industry. The paper appreciates the effect
Calculate the exponential smoothing forecast for weeks 2 through 8 using an initial forecast of 5200 and an alpha of 0.10.
Using MINITAB run the multiple regression analysis using the variables CALLS, TIME, and YEARS to predict SALES. State the equation for this multiple regression model.
In this particular case, Randy will need to assign the correct numbers to the correct category. For the purposes of this case study, assume T will equal 1 to make the equation represent one year of employment in one of the ice cream shops. For following variables, Nn will equal 50 as there will be 50 applicants total selected to be hired, rxy will represent .30 in one equation representing the interview and job performance and in the other equation, it will represent .50 which will represent the work sample predictor and job performance, SDy will be chosen to represent .20, Ẑs will be .80 because it will be the predictor score of the selected applicants, Na will represent 100, as that is the total number of applicants that submitted applications, and Cy will represent the cost per applicant in the interview and job performance in one equation as 100 and it will represent 150 in the other equation for work sample and job
The Standard Error of the Estimate is a quite significant portion of the possible predicted values
1. A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do so, the company randomly chooses 6 small cities and offers the candy bar at different prices. Using the candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below:
* As stated in the guidelines, we also assume that the mean of the demand is equal to the product of the mean of the forecasting error and the forecast itself, and the same for the standard deviation of demand;
Assuming that the demand and supply for premium coffees are in equilibrium, the price will be at a constant, without significant pressure from the market. If Starbucks introduced the world to premium blends, this would cause a positive shift in the demand curve. There a higher equilibrium price and higher quantity when demand increases and supply remain unchanged. As prices increase, and the market moves to a new equilibrium, we will see higher wages, more advances and investments in technology and infrastructure, and greater competition. As production become more efficient and competition becomes greater, supply will increase and cause prices to settle back down. There are several factors that will impact the long-term equilibrium, such as changes in supply. For example, if a hard freeze eliminated Brazil’s premium coffee crop, this would cause a negative shift in the supply curve. Assuming demand remains constant a negative shift in the supply curve will cause quantity to decrease and equilibrium price to increase. Research shows that in 2011 a frost occurred in Brazil's southeastern coffee growing belt. Traders worried that next year's yields could be hurt. At the same time, heavy rains during harvest forced Columbia to reduce its crop estimate for 2011. Understanding the impact of problems along the supply chain and how the changes in supply