Spatial Thinking and Quantitative Methods in Geography

GEOG 2027 Spatial Thinking and Quantitative Methods in Geography

Lab 2 Probability, Distributions and Means Comparison

Due: October 25, 2020

Objectives

You will work on standard normal probabilityin Section 1. In Section 2 you will create hypotheses base on a few research questions. Sections 3concentrates on the use of the Student’s t test and ANOVA to assess the differences between sample means (and by inference, the samples as a whole), and their parent populations.

Data

Start Drive File Stream app and follow screen instructions to start your Google Drive (if you cannot find your Google Drive). Create a Lab2 folder in GEOG2027 on Google Drive. Download Lab2.xlsx and save it to your Lab2 folder for this course.

SECTION 1 PROBABILITY AND DISTRIBUTIONS

Part-1 Standard Normal Distribution

Use EXCEL to create a table of the cumulative probability of the standard normal distribution. Follow the instructions below, and you should start to see what we mean by ‘the cumulative probability’. Note that a ‘standard normal distribution’ is a normal distribution with a mean of zero and a standard deviation of one (i.e. a normal distribution expressed as z-scores).

Start EXCEL and open Lab2.xlsx.Click  at the bottom of the Excel window to add a blank EXCEL worksheet.Create a series in column A that goes from –3.0 to +3.0 (representing standard deviations from the mean of zero) in increments of 0.10:

• Enter z in cell A1
• Enter -3 in cell A2
• Enter =A2+0.1 in cell A3
• Click to select cell A3, place your mouse on the black square at lower right corner of the cell and drag to copy formulae in A3 down to cell 62.

In column B, calculate the cumulative probability of the standard normal deviation.

• Enter p in cell B1
• Opposite the first Z-value in column A (cell B2), use the ‘NORM.S.DIST’ function to calculate the cumulative probability: = NORM.S.DIST (A2, true). You need to double click to select True: cumulative distribution function. Note the “cumulative setting” for Norm.S.Dist. Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, NORMS.DIST returns the cumulative distribution function; if FALSE, it returns the probability mass function. The cumulative distribution function is the probability that the variable takes a value less than or equal to x. Aprobability mass functionis afunctionthat gives the probability that a z value is exactly equal to some value.
• Copy the formulae to all other corresponding cells. The p value should approach (i.e., very close to) 1 as z approaches 3.

You now have a Z table you can refer to.

You calculated Z-score in Lab1. Copy temperature data for Oxford to a blank space in Sheet 1 of your Lab2 file (i.e., do not cover the Z table). Click button on the toolbar to change the number format of calculated z values to two decimal places.Make sure you have mean and standard deviation also calculated in the sheet. Now use the NORM.S.DISTfunction to calculate the cumulative probability associated with each of the z-scores, in the neighboring cells of the column to the right.Follow Example 2 in the lecture to answer following questions.

Q1. What is the probability that temperature will fall below 22°C in August? What is the probability that temperature will fall above 31°C in August?Please show steps of your calculation. (10 points)

Confidence intervals for the mean

If you do not have the Data Analysisbuttonon the Data tab, you need to install the Analysis ToolPak by selectingFile \ Options \ Add-Ins, Click Go… button beside Manage: Excel Add-ins. Check Analysis ToolPak and click OK.

Part 1: The Central Limit Theorem

We will use EXCEL to illustrate the concept of the Central Limit Theorem, and how this relates to the standard error of the mean. This in turn will introduce us to a few new EXCEL capabilities such as the generation of random series.

Add a blank EXCEL worksheet. Start the Data Analysis wizard from the Data tab.Select the Random Number Generation function. Select Normal as the type of distribution. Your data should have 50 variables and 100 random numbers in each series. Leave the mean and SD at 0 and 1 respectively, and the Random Seed blank. You can output the data to a new worksheet. Click OK.

EXCEL may take a few seconds to produce all these numbers. You now have 50 columns (variables) and 100 rows (samples).

1. Calculate the mean for each column, using the appropriate formula. HINT: enter the formula for the first column in a cell beneath the column, then copy the formula across underneath all other columns.
2. Add a blank EXCEL worksheet. For convenience, select the row of 50 means, right click on the selection and selection Copy. Go to thenew worksheet.Click A1 and selectPaste Special in the Paste Specialdialog, select the boxes Values radio button under Paste and check Transpose at the bottom. Click OK.You now have a column (of 50 rows) in the new worksheet: NOTE: make sure you understand what the ‘Values’ and ‘Transpose’ options do.
3. Construct a histogram of the data using the methods employed in Lab1:
   • Bin size: 0.1
   • Bin range: about –1.0 to 1.0
   • Use the histogram tool from the Data Analysis wizard to calculate the frequency distribution
4. Use the chart wizard to plot your frequency histograms.

Repeat steps 1-4, this time using the mean of only the first 10 cells in each column. Copy the row of new means to a new worksheet so that you don’t overwrite the previous work.

Use EXCEL to calculate the mean and standard deviation of the column of 50 means you previously copied from the worksheet with the random numbers.

Q2 Please insert two histogram you created to your Word document. The histograms you produced should qualitatively illustrate the sampling distribution of the means, for two different sample sizes. Please observe the difference of your two histograms and explain how the sample size affects the resultant sampling distribution, and therefore the standard error of the mean. (10 points)

Q3 Use the appropriate formula from thelecture slides to calculate the standard error of the mean, for a population with a mean of zero and standard deviation of one, and a sample size of 100 and 10. (10 points)

A note on EXCEL’s random number generation function: we are using this function for purely illustrative purposes. It should not be used for any application where the results are important to experimental outcome, as the process used by EXCEL is not perfectly random.

Part-2: Confidence intervals around sample means

You have data from a survey of annual income (before tax) from 50 households in North Oxford. The mean income of the households is $50,000, and the standard deviation is $7,500. Using EXCEL, calculate the 95% confidence interval (CI) for the mean.Please calculate the confidence interval following the example from lecture slides first.Then conduct the following:

1. Enter above information, with suitable labels (or headings), into a new worksheet.
2. In a blank cell, enter the confidence levelat which you wish to perform the test (0.95), and label this cell.
3. Enter a formula in another cell that calculates the significance level (α=1-confidence level) associated with your confidence level, and label this as well.
4. Click on a blank celland then click Insert FunctionSelectConfidence.Norm from the Statistics subsection of Or select a category dropdown list and click OK. Alternatively, type in Confidence in then blank at the top of the dialog and then click Go.
   • For Alpha enter the cell reference where you calculated the significance level. Alternatively, you can simply click on and then the cell you want. Click to go back to the dialog. The reference appears in the dialog now;
   • For Standard_deventer the cell reference where you entered the standard deviation of household income;
   • For Size enter the cell reference for the sample size;
   • Is your calculation very close to that of Confidence.norm? NOTE: you could enter actual numbers into the fields of the function wizard, but this doesn’t allow you to do the manipulations in 5 and 6 below.
5. Type a different confidence level (0.99) into the relevant cell and see how the CI changes.
6. Type different sample sizes and standard deviations and once again see how the CI changes.

Save your work.

Confidence intervals for small sample means

Let’s assume that you were short of time, so could only obtain income data for ten households in North Oxford. A preliminary analysis yielded the following statistics:

: $ 45,000, s: $ 8,200, n: 10

Q4 Calculate the 95% confidence interval for the mean of your sample data using Confidence.t function. (10 points)

SECTION 2 HYPOTHESES

Q5 Provide one null and one alternative (research) hypotheses and relevantmathematical expressionsfor the hypotheses for each of the following: (20 points)

a The amount of money spent on food by undergraduate student- athletes and other undergraduates

1. The crime rates in downtown and suburban areas of Toronto
2. The time to complete a task using Method 1 and Method 2
3. The effect of slope aspect on soil temperature in middle and high latitudes

You can stop now. Following sections will be finished next week.

SECTION 3 USING MICROSOFT EXCEL TO CONDUCT T-TEST AND ANOVA

Part 1: t-test

An agricultural research station has been investigating the effects of a new fertilizer on soil cation exchange capacity (CEC). CEC is measured in milli-equivalents (m.e.) of absorbed cations per 100g. A number of random soil samples were taken in two adjacent fields, one of which had been treated using the new fertilizer. The results of the lab work are contained in CECworksheet of Lab2.xlsx. Double click to open this file and click on CEC worksheet.

You will conduct an F-test first to examine the equal variance assumption. Click Data \ Data Analysis. Select F-test two sample for variances and click OK. Choose the ranges for variables 1 and 2. Check Labels. The α value (the significance level of 5%) will stay as 0.05. Set the Output Range as an empty cell (e.g., J1). Click OK.

Q6Mention the null and alternative hypotheses. Insert the F-test output to your Word document. Will you accept or reject the hypothesis? Why? (20 points)

Decide which t-test (Assuming Equal Variances or Unequal Variances) you will use based on the F-test output. Select the appropriate t-test from Analysis Toolpak. Check Labels. Select an Alpha value of 0.05. After specifying the input range (the data) and output ranges (a free cell on your worksheet), click on OK and a table of results will appear.

Read through the labels for the output table and make sure you understand each item, especially the ‘P(T<=t)’ and ‘t Critical’ values for one and two-tailed tests.

Q7: Insert the result (table) to your Word document. Using the information in the EXCEL output table to decide whether you accept or reject a null hypothesis of no difference between the means at the 95% confidence level. Explain. (10 points)

Ask for help if you cannot interpret the table.

Part-2 ANOVA

A researcher wanted to examine whether distance traveled to work varies by income. Eleven individuals in each of three income groups are surveyed. The results are in Distanceworksheet.

You will use ANOVA single factor to do the analysis. Set the α value as 0.05.

Q8Describeyour null and alternative hypotheses of your test. Will you accept or reject the null hypothesis? What conclusion can you make about the differences in travel distance? What isthe probability of mistakenly rejecting the null hypotheses however there are no differences between these groups? (10 points)

Q9There are mathematical relationships between some ANOVA table components (values), for example H=A+B. Examine the ANOVA table and finish three cells with = signs in below table. Do some calculations using your ANOVA table if necessary. For example, Total SS (H) is equal to Between Groups (A) and Within Groups SSs (B).(10 points)