Project #24213 - Statistics Exam


Your state’s spreadsheet provides you with the information that you need to make interpretations, assess the likelihood of various events, select and apply appropriate statistical tests to analyze the data, determine reasonable inferences and predictions to make appropriate decisions.  Problem 1 is worth five points and Problems 2-11 are each worth ten points.  Thus there are a total of 105 points that can be earned.  Five of those 105 points are considered extra credit.  Please submit your completed project via your assignment folder.  The due date is Sunday, 9 March 2014, 23:59 pm EST (Eastern Standard Time).  No late work will be accepted, so consider turning it in early.
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1.  For the frequency distribution of the prices, which is displayed in Part 1 of your spreadsheet, complete the following paragraph by filling in the blanks with appropriate values based upon that frequency distribution. Note that there is more than one correct answer per blank.  You may use any appropriate values that you like as long as they make the paragraph correspond to what is shown in the frequency distribution.

 

            Considering buying a home at a reasonable price?  Then consider      *        .

Approximately _____% of homes surveyed in     *     that are listed on forsalebyowner.com are priced between $_______________ and $______________.  Furthermore, ________% are priced at most $_______________.

 

*Please replace all asterisk blanks with your state’s name.

 

 

 

 

2.  Fill in the ten blanks of the descriptive analysis below using what is displayed in Parts 2 to 5 of the spreadsheet.  The first five, and the 7th, of the ten blanks should be completed with the appropriate numerical values that are already computed for you on the spreadsheet; no further computations should need to be made for the first five and 7th blanks.  When completing the blanks, please pay attention to the footnotes that appear below the following paragraphs but above Problem #3.

 

Considering buying a home at a reasonable price?  Then consider     *     .

 

The average mean price of a home for sale in    *   is $________________, and half of the homes for sale are priced under $__________________.  The most prevalent price of a home is $______________.1 Also it is worth noting that 25% of the advertised prices of homes for sale are under the price of $__________________.

 

In a random sample of homes advertised for sale in     *     on www.forsalebyowner.com, the most expensive home was only $______________ more than the lowest priced home, which means that those desiring a more pricey home can find one here.  However, based upon the _____________________________2 shape of the distribution of prices, there are not many pricey homes for sale in     *    .   Further support of this is given by the coefficient of skewness, which is approximately ________3, and thus implies that the skewness _______________4 significant.  Moreover, the highest home price contained in the homes sampled is _________________________5 high based upon a comparison of it to the price that would be 1.5 times the interquartile range (IQR) above the third quartile and the price that would be 3 times the IQR above the seventy-fifth percentile of the prices. 

 

With respect to the preceding paragraph, it might appear that the prices of the homes advertised for sale in     *      are volatile.  This leads us to factor in the size of the homes into the equation.  In a comparison of the dispersion in the prices, in dollars, of the homes to the sizes, in square feet, of those homes sampled, it can be stated that there is ____________6 variability in the prices than in the sizes of these homes. 

 

 

Footnotes:

 

*Please replace all asterisk blanks with your state’s name throughout this project.

 

1If N/A appears here then an estimate of the most prevalent price of a home could not be determined from the sample taken.

 

2Page 59 of Bluman's Elementary Statistics textbook, 8th edition, was used as a reference for determining a name for the shape.  If the word 'unusual' appears in this blank then the shape of the price data did not reasonably resemble one in this textbook that was used as a reference.

 

3The numerical value selected from the spreadsheet for this blank was the one that best supported the shape stated in the previous blank.

 

4The words 'to the right is' or 'to the right is not' or 'to the left is' or 'to the left is not' written here can be justified by comparing the numerical value of the coefficient of skewness to the numbers 1 and –1.

 

5The selection of the word 'negligently' or 'mildly' or 'extremely' is supported by the numerical computations made in Part 4 of the spreadsheet.

 

6The 'more' or 'less' or 'no real difference in the' that appears in this blank can be justified by computing the coefficients of variation using the sample means and standard deviations shown in Part 3 of the spreadsheet.

  

 

 

 

3.  Each blank below should be completed with a numerical value.  Please use the appropriate numerical values from Part 6 of your spreadsheet to make the computations.  For the first four blanks, which request computed probabilities, please state your responses in unreduced fractional form so that if you have an error I will be able to see what it is and thus I will be better able to explain what your mistake is.  If in addition to the unreduced fractional answer you would also like to state your answer as a reduced fraction or in decimal form or percent form (probably the most common way to state a probability), that would be fine.

 

 

Continuing with our analysis of the prices and sizes of homes advertised for sale in     *    , we should consider the following percentages/probabilities:

·         The likelihood of a home being priced over $300,000 and being less than 2000 square feet in size is _________________.

·         The probability of a home having less than 2000 square feet given that the price is over $300,000 is ____________________.

·         Based upon the random sample of homes taken, the probability of a home being priced from $150,000 to $300,000 or having a size of at least 2000 square feet is _______________________.

·         It can also be stated that the probability of a home not being priced over $300,000 is _______________________.

And finally, if a realtor was contracted to sell the homes that comprise this sample, then the number of ways that 3 of these homes can be selected for a publicity event is ______________.

 

 

 

4.  Each blank below should be completed by a numerical value, rounded to the nearest thousandth.  Let X be the number of homes that use natural gas, and n be the number of 'reasonably priced' homes sampled from       *       . (The value of n to use is shown in Part 11 of your spreadsheet).    

 

Natural gas is used in approximately 60% of American homes.  Source:  Discovery Channel's Curiosity.com.  Assuming this 60% figure applies to       *      7, then in the sample of homes from this state we can expect _________, to use natural gas.  Also the standard deviation of this binomial distribution is ____________.  Furthermore, it might be of interest to note that the probability of exactly 20 of the homes in this sample using natural gas is approximately _______________.

 

7 Note though that in actuality this percentage should apply to a random selection of homes throughout the USA and that the percentage for individual states might vary somewhat from this figure of 60%.

 

 

5.  One of the reasons we might have suspected the original price data is not from a normal bell-shaped distribution is because of the following two comparisons.

 

If the price data that was collected for __*__ was from a normal bell-shaped distribution with a mean m approximately equal to the sample mean shown in Part 3 of the spreadsheet and a standard deviation s that is approximately equal to the sample standard deviation shown in Part 3 of the spreadsheet, then the expected probability of a randomly selected home having a price greater than $300,000 (symbolized P(x > 300,000)) would be _________%.  However, based upon the sample data that was collected, the actual percentage of the homes with a price greater than $300,000 is _________%, which differs from what we would expect, if the prices were normally distributed, by _____________8.

 

Furthermore, based upon a normal distribution with a mean m and a standard deviation s that are respectively approximately equal to the sample mean and sample standard deviation that are shown for the price data in Part 3 of the spreadsheet for __*___, we would expect 2/3 or about 67% of the homes to be priced under $_______________.  However, using the actual data collected the percentage of homes priced under that amount (the dollar amount shown in the non-asterisk blank of the previous sentence) could be computed by counting how many of the homes sampled have prices below the aforementioned price and then dividing by the number of homes sampled.  Most likely that percentage will not be approximately 67%.  (See Part 15 of spreadsheet.)

 

8 The number shown in this blank was calculated by finding the difference between the two percentages shown in the preceding two blanks.

 

 

 

6.  Use your original price data shown in the chart in Part 6 to help you correctly complete the following blanks.  You will also need the computation made for the margin of error shown in Part 14 of your spreadsheet to help you complete the second and third blanks.

 

Based upon the sample of __*__ home prices collected, it was found that _______% of them are over $300,000.  Using that percentage, it can be stated with 90% confidence that the true proportion of advertised selling prices for all homes for sale by owner in __*__  that are over $300,000 is between _________% and ________%.  However, if we needed to be 95% confident that the sample estimate of the population proportion of all homes advertised for sale by owner in ___*___ that are priced over $300,000 differs by no more than 5% from the true proportion then we would need to collect a sample of at least _________9 home prices. 

 

9 This blank should be completed with an estimate of the minimum sample size needed.

 

 

 

7.  Refer to Parts 7 and 11 of your spreadsheet.

 

Let's say the       *       Times reported that the average advertised (reasonable) selling price on forsalebyowner.com for a home in ___*___  is less than $_________. (Replace this blank with the value shown in Part 7 Cell i83.)  To determine the validity of this claim we will conduct a formal hypothesis test using the trimmed data that produced the sample statistics shown in Part 11 of the spreadsheet for ____*____.  The statement of hypothesis is Ho:________________  versus H1:____________________. Using a 10% significance level, the critical value is found to be _________.  Comparing the test statistic value shown in Part 7 of the spreadsheet to that critical value we arrive at the decision to _____________10 the null hypothesis Ho and conclude that the average advertised (reasonable) selling price on forsalebyowner.com for a home in this state __________11 significantly less than the amount claimed by the ___*___       Times.

 

10This blank should be completed with either ‘reject’ or ‘fail to reject’ based upon the comparison you make.

 

11This blank should be completed with either 'is' or 'is not' based upon the decision regarding Ho stated in the previous blank.

 

 

 

8.  Assume that the population distribution of trimmed prices for both      _____*_____ and Idaho are normally distributed.  Refer to Part 11 of your state’s spreadsheet for any sample values needed for ___*___.  Some sample values from Part 11 of Idaho’s spreadsheet are:

n = 31,  sample mean = $210,719.32,  s = $67,563.43

 

A realtor claimed that there is a significant difference in the average advertised selling prices of homes in       *       from that in Idaho. Considering all the homes listed for sale in       *       and all the homes listed for sale in Idaho we want to determine if the population means of the advertised selling prices of all(reasonably priced) homes in these two states could be assumed to be sufficiently differentThe statement of hypothesis is Ho:________________  versus H1:__________________.  The appropriate distribution to use for this test of hypothesis is the _____________12, and the computed test statistic value is __________.  Using a significance level of 0.05, the critical value(s) is/are _________, and the appropriate decision is to ______________13 the null hypothesis.  Thus it can be concluded that the realtor's claim __________14 supported.

 

12This blank should be completed with one of the following as appropriate:  chi-square, F, t, z-standard normal.

 

13This blank should be completed with either ‘reject’ or ‘fail to reject’ as appropriate.

 

14This blank should be completed with either 'is' or 'is not' based upon the decision regarding Ho stated in the preceding blank.

 

 

 

9.   Please view Parts 8 and 9 of your spreadsheet when answering the following regarding the advertised prices of homes for sale         in     *     and the corresponding sizes, in square feet, of those homes.

 

a)  Compare the value of the correlation coefficient, r, to the appropriate critical values in Table I of the appendix of our textbook (or other PPMC table).  There is a ____________.  Fill in the blank with the best answer from i to iv stated below.

                          i.     significant linear relationship between the advertised prices of homes for sale and the corresponding sizes of the homes at the 5% significance level only

 

                        ii.     significant linear relationship between the advertised prices of homes for sale and the corresponding sizes of the homes at the 1% level of significance only

 

                       iii.     significant linear relationship between the advertised prices of homes for sale and the corresponding sizes of the homes at both the 5% and the 1% levels of significance

 

                        iv.     significant linear relationship between the advertised prices of homes for sale and the corresponding sizes of the homes at neither the 5% nor the 1% levels of significance

 

b)  In    *    if a home has an advertised sale price of $250,000, then the predicted size of the home is ___________ square feet.

 

 

 

10.Let’s say that the government stated that in the USA 30% of homebuyers are first-time buyers, 50% of homebuyers have purchased exactly one home prior to their current acquisition, and 20% of homebuyers have previously purchased 2 or more homes.  You suspect these percentages are different for    *   .  You will plan to test your claim using a 0.05 level of significance.  The statement of hypothesis is Ho:________________  versus H1:__________________.  The appropriate distribution to use to test the claim is the ______________15.  This is a __________-tailed test.16  The appropriate test statistic formula is the one found on page number _______17 of our textbook.  If the p-value was found to be 0.043, then the appropriate decision should be to ______________18 the null hypothesis. 

 

15This blank should be completed with one of the following as appropriate:  chi-square, F, t, z-standard normal.

 

16This blank should be completed with one of the following as appropriate: left, right, two.

 

17This blank could alternatively be completed by stating the formula rather than the page number.

 

18This blank should be completed with either ‘reject’ or ‘fail to reject’ as appropriate.

 

 

 

11.We want to test the hypothesis, using a=0.05, that there is no significant difference in the average square footage of all homes listed for sale by owner in the states Alaska, Nevada, Rhode Island, Wyoming, and       *       that are advertised on forsalebyowner.com.

Here are the respective sample sizes that were taken/obtained for each of those states:  34, 40, 32, 30, and ____ (this blank should be completed with the value of n shown in Part 3 of your spreadsheet).

Assume that each state's distribution of sizes in square feet is normally distributed.  Provide an appropriate response to each of the following.

 

a)  The statement of hypothesis is Ho:________________  versus H1:__________________.

 

b)  Complete the ANOVA table below replacing all seven question marks with the correct values.

 

Analysis of Variance Summary Table

 

 

 

        Sum of

 

         Mean

 

Source

       Squares

       d.f.

        Square

            F

Between

11.2

        ?

              ?

            ?

Within(error)

?

        ?

              ?

 

   Total

217

        ?

 

 

 

 

c)  The critical value from the F-distribution table, Table H in our textbook's appendix, using a 0.05 significance level and d.f.Numerator=k-1 and d.f.Denominator=N-k, is somewhere between 2.45 and 2.37.  Using 2.37 for the critical value, the appropriate decision is to ______________19 the null hypothesis.  AND, which of the following can be concluded?

 

·         There is no significant difference in the average square footage of all homes listed for sale by owner in the states Alaska, Nevada, Rhode Island, Wyoming, and       *       that are advertised on forsalebyowner.com.

 

·         There is a significant difference in at least one of the average square footages of all homes listed for sale by owner in the states Alaska, Nevada, Rhode Island, Wyoming, and __*___      that are advertised on forsalebyowner.com from the other average square footages of homes advertised for sale in those states.

 

·         There is a significant difference in all of the average square footages of all homes listed for sale by owner in the states Alaska, Nevada, Rhode Island, Wyoming, and       *       that are advertised on forsalebyowner.com.  In other words, each of those states has a mean square footage that is significantly different from the other four states in this list.

 

19This blank should be completed with either ‘reject’ or ‘accept/fail to reject’ as appropriate.

Subject Mathematics
Due By (Pacific Time) 03/05/2014 07:00 pm
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