/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 The following table shows Apple ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 39

The following table shows Apple iPhone sales from the 2nd quarter in 2007 through the second quarter in \(2008(t=2\) represents the second quarter of 2007\():^{.11}\) $$ \begin{array}{|r|c|c|c|c|c|} \hline \text { Quarter } \boldsymbol{t} & 2 & 3 & 4 & 5 & 6 \\ \hline \begin{array}{r} \text { iPhone Sales } \\ \text { (thousands) } \end{array} & 270 & 1,119 & 2,315 & 1,703 & 717 \\ \hline \end{array} $$ a. Find a quadratic regression model for these data. (Round coefficients to the nearest whole number.) Graph the model together with the data. b. What does the model predict for iPhone sales in the third quarter of \(2008(t=7)\) to the nearest 1,000 units? Comment on the answer, and ascertain the actual third quarter sales in 2008 ( Apple's fiscal fourth quarter).

Short Answer

Expert verified
The quadratic regression model for the given Apple iPhone sales data is approximately \(y = -196x^2 + 2816x - 3815\), where \(x\) represents the quarter and \(y\) represents the sales in thousands of units. Using this model, the predicted iPhone sales for the third quarter of 2008 (t=7) are approximately 1,000 units. However, actual sales in the third quarter of 2008 were significantly higher, which may be attributed to factors not accounted for in the model, such as new product releases or market conditions.

Step by step solution

01

Input sales data into a graphing calculator

Enter the given sales data as (quarter, sales) pairs. For example, enter (2, 270), (3, 1119), (4, 2315), (5, 1703), and (6, 717).
02

Find the quadratic regression model for the data

In the graphing calculator, go to the Regression menu and choose the Quadratic Regression option. The calculator will give you the quadratic model equation in the form \(y = ax^2 + bx + c\), where a, b, and c are the coefficients of the equation. Round the coefficients to the nearest whole number.
03

Graph the model together with the data

Once you have the quadratic model equation, plot it on a graph along with the original sales data. You should see the curve of the model is a good fit to the data.
04

Predict the sales in the third quarter of 2008

To predict the sales in the third quarter of 2008 (t=7), plug t=7 into the quadratic model equation. Calculate the estimated sales by rounding the result to the nearest 1,000 units.
05

Analyze the prediction

Based on the quadratic model equation, discuss the accuracy of the prediction, taking into account any factors that might have influenced the sales in the third quarter of 2008. To check the accuracy of the prediction, find the actual third quarter sales in 2008 (Apple's fiscal fourth quarter). Compare the actual sales to the predicted sales and discuss the possible reasons for any differences.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

iPhone Sales Data
When analyzing iPhone sales data, it's important to understand the trend from the second quarter of 2007 through the second quarter of 2008. This time frame represents the early growth phase of iPhone sales. The data points to significant increases in sales, peaking in the fourth quarter at 2,315,000 units. The growth is followed by a decline, dropping to 717,000 units in the sixth quarter.

This pattern suggests fluctuations often present in the technology market. Factors attributing to sales spikes can include product launches, marketing campaigns, or market expansion.
  • Quarter 2 (2007): 270,000 units
  • Quarter 3: 1,119,000 units
  • Quarter 4: 2,315,000 units
  • Quarter 5: 1,703,000 units
  • Quarter 6: 717,000 units
Seeing these numbers through the lens of quadratic regression helps to understand and possibly predict future trends.
Graphing Calculator Steps
To find a quadratic regression model, you can use a graphing calculator. Here's how you can achieve this using simple steps. Begin by entering the provided sales data as pairs consisting of the quarter number and sales figure, like (2, 270) and so forth, until (6, 717).

After entering the data:
  • Access the Regression menu on your graphing calculator.
  • Select the 'Quadratic Regression' option, which calculates the best-fit curve in the form of the equation \(y = ax^2 + bx + c\).
  • Review the resulting coefficients \(a\), \(b\), and \(c\), and round them to the nearest whole number.
Next, proceed to visually check how well the model fits the data by plotting both the calculated model and the real sales data. A good fit typically means the curve closely aligns with the data points. This process illustrates how historical data can inform on the future trends.
Sales Prediction Analysis
Predicting future sales involves applying the obtained quadratic regression model. For instance, to estimate the iPhone sales in the third quarter of 2008 (\(t = 7\)):

  • Substitute \(t = 7\) into the quadratic equation \(y = ax^2 + bx + c\).
  • Calculate the estimated sales and round to the nearest 1,000 units for simplicity.
Once you have the predicted value, compare it against the actual sales. When comparing forecasts to real outcomes, it's crucial to consider external influences like economic conditions, competitor actions, or product updates that the model does not account for.

This predictive operation helps in realizing how trends might shift and allows companies to strategize accordingly, potentially mitigating risks or leveraging positive trends.

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