statistics-and-probability-number-2-having-convinced-the-board-members

his is the Second question

Number 2. Having convinced the board members that a stratified sample is the most appropriate, you collect data from the 80 schools. The spreadsheed provided contains the collected data in the tab labelled “”Data.”” The difference between average school test scores three years ago and average school test scores today is recorded as “”chgtestscores.”” Positive values of chgtestscores indicate an increase in test scores at the school as compared to 3 years ago, while a negative number indicates that the school is now performing worse on these tests. In addition to this variable, there are three more variables in this data set. The first is “”curriculum”” which can have a value of either “”old”” or “”new,”” where new implies the experimental curriculum. Next is “”income(\$000),”” which represents the average annual income (in thousands of dollars) of the households of students from each school. Finally, there is a variable called “”school”” which is simply ID number of the elementary school. The first step to this analysis is to generate some descriptive measures.

For each of the following points, create the chart and/or graph that best displays the data:
a) Show the breakdown of your sample by curriculum type (old vs. new/experimental)
b) Show the distribution of the change in test scores across all schools.
c) Show the distribution of income across all schools.

Additionally, you want to generate some tables of summary statistics.
d) Create one table that calculates summary measures of the change in test scores and income variables across all 80 schools.
e) Create a second table that calculates summary measures of the change in test scores and income variables broken out by curriculum type.

Based on the graphs and tables created in parts a-e:
f) What preliminary conclusions can you draw regarding the effectiveness of the experimental curriculum?”

For this exam, assume that you are working on a team that has been commissioned by a large school district to
collect and analyze data related to a recent curriculum experiment designed to improve student scores on state-wide
standardized tests. The schools in this district are predominantly large, urban schools. They are interested in
knowing how successful the experiment was and if the new curriculum should be incorporated district-wide. Three
years ago, the school district rolled out the experimental curriculum to 100 of the 400 elementary schools in the
district. Those 100 schools were selected via a simple random sample. Your working budget is not large enough to
collect data on the population of 400 schools and you can afford to only collect a sample of 80 schools. Unless the
question states otherwise, conduct all analyses at the 95% confidence level (α=.05).

1. You think that the best sampling strategy is stratified sampling. You’d like to list the characteristics of schools in
this district, then randomly select 80 schools who roughly match the demographic characteristics of the entire
population of schools in this district. 40 of these would come from schools that had the experimental curriculum and
40 would come from schools that kept the old curriculum.
However, people from the school board have made the following statements regarding sampling. One well meaning
school board member has argued that &quot;If it ain’t broken, there’s no reason to fix it! We should sample 80 random
schools with the old curriculum; as long as the students in those schools are performing acceptably, there is no
reason to change curricula.&quot; A second school board member who represents the teachers union has expressed
concerns that &quot;these results will be used as evidence against poor teachers or schools and that is unfair to them. You
should only look at the subset of schools that improve to protect the identies of those who work at the
underperforming schools.&quot; Finally The school board member who authored the new curriculum has told you &quot;look, to
be honest, the old curriculum was horrible compared to the new one, but many teachers and parents are simply
reluctant to change. The Secretary of Education testified in the US Congress that ‘this curriculum is quite honestly
the best curriculum in the country. It would be foolish not to roll it out to all schools immediately.’ The best thing for
our kids is to have the best performing schools with the new curriculum and the worst performing schools with the old
curriculum in the sample…the bigger we can make the impact look, the greater the support will be to adopt the new
curriculum district wide.&quot;
Describe what is wrong methodologically with each of the three suggestions you received from the various school
board members. You will want to focus on issues such as sampling error, the sorts of biases that will be introduced
from such sampling methods, how you might expect to see those biases manifested in the data/data analysis, and
how these issues will affect your ability to comment on the district’s original question as to whether or not it would be
a good idea to roll out the new curriculum to all schools in the district.

2. Having convinced the board members that a stratified sample is the most appropriate, you collect data from the 80
schools. The spreadsheed provided contains the collected data in the tab labelled &quot;Data.&quot; The difference between
average school test scores three years ago and average school test scores today is recorded as &quot;chgtestscores.&quot;
Positive values of chgtestscores indicate an increase in test scores at the school as compared to 3 years ago, while a
negative number indicates that the school is now performing worse on these tests. In addition to this variable, there
are three more variables in this data set. The first is &quot;curriculum&quot; which can have a value of either &quot;old&quot; or &quot;new,&quot;
where new implies the experimental curriculum. Next is &quot;income(\$000),&quot; which represents the average annual
income (in thousands of dollars) of the households of students from each school. Finally, there is a variable called
&quot;school&quot; which is simply ID number of the elementary school. The first step to this analysis is to generate some
descriptive measures.
For each of the following points, create the chart and/or graph that best displays the data:
a) Show the breakdown of your sample by curriculum type (old vs. new/experimental)
b) Show the distribution of the change in test scores across all schools.
c) Show the distribution of income across all schools.
Additionally, you want to generate some tables of summary statistics.
d) Create one table that calculates summary measures of the change in test scores and income variables across all
80 schools.
e) Create a second table that calculates summary measures of the change in test scores and income variables
broken out by curriculum type.
Based on the graphs and tables created in parts a-e:
f) What preliminary conclusions can you draw regarding the effectiveness of the experimental curriculum?

3. One of the criticisms levied upon the the old curriculum is that it was outdated. It was so outdated, the board
members argue, that it was causing standardized test scores to fall. You decide to test this hypothesis.
a) State the null and alternative hypotheses (H0 and H1)
b)Test the hypothesis that the mean change in schools using the old curriculum was less than 0.
c) Calculate the p-value associated with your test statistic from b.

4. Because the school board’s primary concern is whether or not the experimental curriculum led to better
standardized test scores, your next step is to conduct a simple analysis comparing test scores from schools with the
old curriculum with the test scores from schools with the new/experimental curriculum.
a) Conduct an ANOVA to evaluate whether or not there is a significant difference in test scores between schools with
the old curriculum and schools with the new curriculum.
b) Summarize and Interpret the results of your test.
5. The board member who originally wanted you to include only low income households in the survey is still
concerned about the particular effect of the experimental curruculum on schools in low income neighborhoods. TO
find the answer to this, you need to run a multiple regression model.
a) Create a dummy variable for the experimental curriculum and an interaction variable that interacts the experimental
dummy and the income variable.
b) Estimate a multiple regression model that includes the curriculum dummy, income, and interaction variable as
independent variables.
c) Calculate predicted values for the chgtestscores variable for both the new and old curriculum for income levels of
\$15,000, \$30,000, \$60,000, and \$120,000.
d) Summarize and interpret the results of this model. What do you tell the board member about the effect of the new
curriculum across different income levels?
6. Shortly after you publish your findings in a report, you receive a call from a small midwestern school district. The
schools in this district are mostly small schools in rural areas and farming communities with very low populations.
Unbeknownst to anyone, they too experimented with the exact same curriculum used in your school district at the
same time. However, they are confused because their statistical findings showed nothing significant. What do you
tell them and why?
7. The variable the school board is most interested in understanding/explaining is the change in schoolwide
standardized test scores. You were also given variables that indicated the curriculum type and the income of the
households of students from each school. If this were a real research project, you would surely collect other data to
use as control variables. For each of the following variables, explain why you would, or would not, want to collect the
data to use in your analysis: a) percentage of students who are ESL (English as a Second Language) students, b)
experience of the school superintendant, c) change in test scores over the three years prior to the experimental
curriculum, d) average change in test scores on a DIFFERENT standardized test, and e) percentage of households
with single parents

chgtestscores curriculum
-0.12 Old
1.63 New
0.59 New
0.86 Old
1.44 New
0.43 Old
0.33 New
-1.47 Old
-0.53 Old
-0.4 New
0.59 Old
-0.55 Old
0.69 New
1.27 New
-0.46 Old
1.11 New
-0.13 New
0.72 New
0.92 Old
-0.08 New
0.13 Old
0.52 Old
0.25 Old
-0.31 New
-0.08 New
-1.14 Old
0.13 Old
-0.07 Old
-0.82 Old
0 Old
0.45 New
2.2 New
1.05 New
2.24 New
-1.54 Old
1.28 New
-0.4 Old
0.51 New
-1.5 Old
-0.49 New
1.16 New
1.43 Old
1.58 New
1.12 New
-0.49 Old
-0.22 New
-0.68 New
1.55 New
-0.11 New
1 New
1.64 New
0.43 Old
-0.16 Old
0.43 New
0.01 New
0.56 Old
1.18 Old
-0.64 Old
0.63 New
-0.05 Old
0.26 Old
-0.39 Old
-0.27 New
-1.1 New
-1.11 Old
-0.62 Old
-0.21 Old
1.12 Old
0.72 New
2.16 New
-0.34 Old
-0.37 Old
1.07 Old
0.24 Old
-0.23 New
0.25 New
-0.42 Old
1.44 Old
1.15 New
0.02 New

income (\$000) school
75.4
3
50.1
11
48.4
12
80.6
16
56.4
17
11.7
23
20
34
85
38
33.3
43
22
50
36.6
53
19.7
65
53.4
67
42.7
69
29.4
70
40.4
83
43.5
85
22.4
88
29.4
89
40.5
92
20.5
94
24.3
101
24.5
105
19.8
118
15.5
120
107.1
130
41.1
131
42.3
141
29.8
142
28.9
145
74.3
152
99.6
155
60
159
68.9
171
101.1
174
36.6
175
29.7
180
46.5
193
92.1
194
34.5
195
61.4
200
10.5
212
71.4
215
27.9
218
62.6
220
27.2
223
15.1
228
111.3
237
40.5
238
28
239
48.2
248
54
259
57.5
261
26.1
262
11.6
272
102.8
274
8.5
275
23.5
289
18.7
291
38.6
294
47.7
296
20.1
303
14
304
17.5
309
41.7
316
93.5
322
23.7
331
32.9
334
22.6
335
76.9
344
126.2
347
22.2
348
6.4
363
49.6
366
45.7
367
19.9
368
30.6
379
27.3
382
68.3
383
23.9
398

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