Math Problem Statement

The price per share of stock for a sample of 25 companies was recorded at the beginning of 2012 and then again at the end of the 1st quarter of 2012. How Activity Frame n during the 1st quarter is an indicator of what is ahead for the stock market and the economy. The sample data are provided in the Excel Online me verow. cunstruct a spreadsheet to answer the following questions. Due to a recent change by Microsoft you will need to open the XLMiner Analysis ToolPak add-in manually from the home ribbon. Screenshot of ToolPak X Open spreadsheet a. Let di denote the change in price per share for company i where di = 1st quarter of 2012 price per share minus the beginning of 2012 price per share. Use the sample mean of these values to estimate the dollar amount a share of stock has changed during the 1st quarter $ 7.64 (to 2 decimals) b. What is the 95% confidence interval estimate of the population mean change in the price per share of stock during the first quarter? Interpret this result. Standard deviation (to 2 decimals): 4.04 Confidence interval (to 2 decimals): ( 6.06 8, 9.22 The mean price per share has increase between 6.06

  • % and 9.22.
  • % over the three-month period

Here is the data:

Set 1: End of 1st Quarter 26.43 39.11 54.60 60.62 67.26 110.28 31.51 13.55 44.41 21.77 33.83 22.95 51.71 47.78 69.81 41.13 72.92 42.97 34.66 76.41 70.66 88.63 25.15 31.66 108.41

Set 2:Beginning of Year 18.01 34.36 43.28 59.13 62.78 103.64 21.61 7.81 38.92 15.21 28.64 16.38 39.74 32.15 64.01 38.02 60.10 39.16 24.86 68.87 51.80 81.86 19.24 23.08 104.60

Solution

Let's go through the steps to answer each part of the question:

a. Estimate the Dollar Amount of Change in Stock Price

We are tasked with calculating the mean change in price per share during the first quarter of 2012, using the formula:

di=Price at the end of 1st Quarter−Price at the beginning of 2012d_i = \text{Price at the end of 1st Quarter} - \text{Price at the beginning of 2012}

First, we'll compute the difference did_i for each of the 25 companies, then find the mean of these changes.

Here are the differences did_i:

26.43 - 18.01 & = 8.42 \\ 39.11 - 34.36 & = 4.75 \\ 54.60 - 43.28 & = 11.32 \\ 60.62 - 59.13 & = 1.49 \\ 67.26 - 62.78 & = 4.48 \\ 110.28 - 103.64 & = 6.64 \\ 31.51 - 21.61 & = 9.90 \\ 13.55 - 7.81 & = 5.74 \\ 44.41 - 38.92 & = 5.49 \\ 21.77 - 15.21 & = 6.56 \\ 33.83 - 28.64 & = 5.19 \\ 22.95 - 16.38 & = 6.57 \\ 51.71 - 39.74 & = 11.97 \\ 47.78 - 32.15 & = 15.63 \\ 69.81 - 64.01 & = 5.80 \\ 41.13 - 38.02 & = 3.11 \\ 72.92 - 60.10 & = 12.82 \\ 42.97 - 39.16 & = 3.81 \\ 34.66 - 24.86 & = 9.80 \\ 76.41 - 68.87 & = 7.54 \\ 70.66 - 51.80 & = 18.86 \\ 88.63 - 81.86 & = 6.77 \\ 25.15 - 19.24 & = 5.91 \\ 31.66 - 23.08 & = 8.58 \\ 108.41 - 104.60 & = 3.81 \\ \end{align*}$$ To calculate the **mean** change: $$\text{Mean change} = \frac{1}{25} \sum_{i=1}^{25} d_i = \frac{190.97}{25} = 7.64$$ So, the mean change is **$7.64**. ### b. 95% Confidence Interval for Population Mean Change The formula for a confidence interval for the mean is: $$\text{CI} = \bar{x} \pm t \left(\frac{s}{\sqrt{n}}\right)$$ Where: - $$\bar{x} = 7.64$$ is the sample mean, - $$s = 4.04$$ is the sample standard deviation, - $$n = 25$$ is the sample size, - $$t$$ is the critical value from the t-distribution for 95% confidence and 24 degrees of freedom. Using a **t-value** of approximately 2.064 for 95% confidence and $$n - 1 = 24$$ degrees of freedom: $$\text{Margin of error} = 2.064 \times \left(\frac{4.04}{\sqrt{25}}\right) = 2.064 \times 0.808 = 1.67$$ Thus, the confidence interval is: $$7.64 \pm 1.67 = (7.64 - 1.67, 7.64 + 1.67) = (5.97, 9.31)$$ So the 95% confidence interval estimate of the population mean change is **(5.97, 9.31)**. ### Interpretation This confidence interval means that we are 95% confident that the true mean change in stock price for the population lies between $5.97 and $9.31 during the first quarter of 2012. --- Would you like more details or have any questions? Here are some related questions: 1. How do we interpret the significance of the confidence interval in financial predictions? 2. What factors could influence the variation in stock prices in this context? 3. How would the confidence interval change if the standard deviation was higher? 4. Why do we use a t-distribution instead of a z-distribution for this sample? 5. How would increasing the sample size affect the confidence interval? **Tip:** When constructing confidence intervals, it's important to ensure the sample size is sufficiently large to provide accurate estimates of the population parameters.

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Confidence Interval
Stock Price Analysis

Formulas

Mean change formula: Mean = (Sum of changes) / n
Confidence interval formula: CI = x̄ ± t(s/√n)

Theorems

Central Limit Theorem
T-distribution for confidence intervals

Suitable Grade Level

Undergraduate (Statistics or Finance)

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