MS 150 Statistics Fall 2000 FX

Part I

Use this infant mortality data to answer the following questions.

Location Infant Mortality
per
1000 live births
American
Samoa		 11
CNMI 38
Fiji 17
FSM 33
Guam 7
Kiribati 55
Marshall
Islands		 41
Nauru 11
Palau 17
Samoa 33
Vanuatu 63
  1. _________ Determine the sample size n.

  2. _________ Calculate the sample mean xbar.gif (842 bytes).

  3. _________ Determine the median.

  4. _________ Determine the mode.

  5. _________ Calculate the range.

  6. _________ Calculate the sample standard deviation sx.

  7. _________ Calculate the sample variance.

  8. _________ Calculate the Coefficient of Variation.

  9. Using the intervals specified in the bins column, fill in the frequency and relative frequency columns of the following table. The number in the bins column is the class upper limit.  Include the class upper limit in each interval.
Bins Frequency Relative
Frequency f/n
10 _________ _________
20 _________ _________
30 _________ _________
40 _________ _________
50 _________ _________
60 _________ _________
70 _________ _________
Sums: _________ _________
  1. Draw a histogram of the Relative Frequency data using the following chart:

 histfx2000.gif (4192 bytes)

  1. _________ What is the shape of the distribution?
  2. _________ What is the probability that, for the Pacific Island locations above, the infant mortality rate will be greater than 10 but less than or equal to 20 per thousand live births?

  3. Construct a 95% confidence interval for the population mean m infant mortality rate per 1000 live births for Pacific Islanders using the above data.  Presume for this example that the data distribution is sufficiently normal.  Note that n is less than 30. Use the sample mean and sample standard deviation to generate your maximal error of estimate.  Show all of your work either below or on the back of this sheet.






    Degrees of freedom = __________

    The maximal Error of Estimate E = _______________

    The 95% confidence interval for m is  ____________ < m < ____________

  4.   In the following exercise use your sample mean and sample standard deviation as population parameters.  That is, use your calculations in #2 above, the sample mean xbar.gif (842 bytes), for the population mean m.  Use your calculations in #6 above, the sample standard deviation sx, for the population standard deviation s.  

    Let the null hypothesis H0 be that the average infant mortality rate is m as calculated by you in question number two above.  Suppose in the year 2000 the actual average infant mortality rate for these same countries is 24.  At an alpha of 0.05, is this drop in the infant mortality rate statistically significantly less than the present rate?

    Do we reject H0 and say the drop is statistically significant or do we fail to reject H0 and say the change is random at a=0.05?  Note that n is less than 30.   You are welcome to calculate a p-value if the power is on and you know how to do this, but it is not required. Show all of your work either here or on the back! 

    1. _______________ What is H0?

    2. _______________ What is H1?

    3. __________ What is a?

    4. __________ What is the t-statistic?

    5. __________ What is t-critical?

    6. __________ Do we reject H0?

    7. __________ What is the p-value? [OPTIONAL!]
Part II

For this part use the following data:

Location Per Capita
Income
in dollars		 Infant Mortality
per
1000 live births
American
Samoa		 3300 11
CNMI 13100 38
Fiji 2000 17
FSM 1900 33
Guam 20700 7
Kiribati 600 55
Marshall
Islands		 1500 41
Nauru 9400 11
Palau 6400 17
Samoa 1300 33
Vanuatu 1276 63
  1. _________ Calculate the slope of the least squares line for the income and infant mortality data above.


  2. What does of the sign of the slope tell us about this data?  That is, what type of correlation is this?


  3. _________ Calculate the linear correlation coefficient r for the income and infant mortality columns.

  4. _________ Is the correlation none, low, moderate, high, or perfect?

  5. _________ Calculate the coefficient of determination.

  6. What does the coefficient of determination tell us about the relationship between per capita income and infant mortality rates?



Statistic Equations Excel
Sample size n =COUNT(data)
Population size N =COUNT(data)
Sample mean xbar.gif (842 bytes) = sigmaxovern.gif (915 bytes) =AVERAGE(data)
=SUM(data)/COUNT(data)
Formulas for the population mean m = x P(x)
= n p
Sample Standard Deviation = sx
=sampstdev.gif (1072 bytes)		 =STDEV(data)
Population Standard Deviation = sigmax.gif (872 bytes)
=probabilitypopstdev.gif (1053 bytes)
= npq.gif (927 bytes)		 =STDEVP(data)
Slope =SLOPE(y data, x data)
Intercept =INTERCEPT(y data, x data)
Correlation =CORREL(y data, x data)
Binomial probability = nCr pr q(n-r) =COMBIN(n,r)*p^r*q^(n-r)
Calculate a z value from an x z = standardize.gif (905 bytes) =STANDARDIZE(x, m, s)
Calculate an x value from a z x = sz+m
Calculate a cumulative probability from a z value where the probability is calculated from negative infinity to z. =NORMSDIST(z)
Calculate a z value from a probability where the probability is calculated from negative infinity to z. =NORMSINV(probability)
Calculate a z value from an xbar.gif (842 bytes) value given m and s xbartoz.gif (1022 bytes) =STANDARDIZE(x, m, s/SQRT(n))
Calculate an error tolerance E for large n (n>30) error_tolerance_e.gif (987 bytes) Excel uses a in the following function where a = 1 - confidence level:
=CONFIDENCE(a,s,n)
Calculate a confidence interval for a mean m for large n (n>30) using the population deviation s confidencelargen.gif (1296 bytes)
Calculate a confidence interval for a mean m for large n using the sample deviation s confidencelargen_s.gif (1260 bytes)
Calculate a confidence interval for a mean m for small n using the sample deviation s confidencesmalln_s.gif (1259 bytes)
Calculate a critical t-value for a two tailed t-distribution [ Excel TINV: If seeking a one tail probability, multiply alpha by two.  I do NOT recommend students use this function at this time, refer to your tables. ] Use table =TINV(level of significance,degrees of freedom)

 
Calculate a critical t-value for a one tailed t-distribution [See above] Use table =TINV(2*level of significance,degrees of freedom)
t-statistic or t-ratio or tdata xbartot.gif (1028 bytes) =STANDARDIZE(xbar.gif (842 bytes), m, s/SQRT(n))
P-value from a t-statistic, degrees of freedom, and number of tails (1 or 2).   Note that the t-statistic must be the absolute value of the t-statistic.   Excel only accepts positive t-statistics. =TDIST(t-statistic,degrees of freedom, number of tails)

normal_curve.jpg (22909 bytes)

Table of standard normal probabilities from 0 to z.  For values of z larger than 2.69 use 0.497.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 0.032 0.036
0.1 0.040 0.044 0.048 0.052 0.056 0.060 0.064 0.067 0.071 0.075
0.2 0.079 0.083 0.087 0.091 0.095 0.099 0.103 0.106 0.110 0.114
0.3 0.118 0.122 0.126 0.129 0.133 0.137 0.141 0.144 0.148 0.152
0.4 0.155 0.159 0.163 0.166 0.170 0.174 0.177 0.181 0.184 0.188
0.5 0.191 0.195 0.198 0.202 0.205 0.209 0.212 0.216 0.219 0.222
0.6 0.226 0.229 0.232 0.236 0.239 0.242 0.245 0.249 0.252 0.255
0.7 0.258 0.261 0.264 0.267 0.270 0.273 0.276 0.279 0.282 0.285
0.8 0.288 0.291 0.294 0.297 0.300 0.302 0.305 0.308 0.311 0.313
0.9 0.316 0.319 0.321 0.324 0.326 0.329 0.331 0.334 0.336 0.339
1.0 0.341 0.344 0.346 0.348 0.351 0.353 0.355 0.358 0.360 0.362
1.1 0.364 0.367 0.369 0.371 0.373 0.375 0.377 0.379 0.381 0.383
1.2 0.385 0.387 0.389 0.391 0.393 0.394 0.396 0.398 0.400 0.401
1.3 0.403 0.405 0.407 0.408 0.410 0.411 0.413 0.415 0.416 0.418
1.4 0.419 0.421 0.422 0.424 0.425 0.426 0.428 0.429 0.431 0.432
1.5 0.433 0.434 0.436 0.437 0.438 0.439 0.441 0.442 0.443 0.444
1.6 0.445 0.446 0.447 0.448 0.449 0.451 0.452 0.453 0.454 0.454
1.7 0.455 0.456 0.457 0.458 0.459 0.460 0.461 0.462 0.462 0.463
1.8 0.464 0.465 0.466 0.466 0.467 0.468 0.469 0.469 0.470 0.471
1.9 0.471 0.472 0.473 0.473 0.474 0.474 0.475 0.476 0.476 0.477
2.0 0.477 0.478 0.478 0.479 0.479 0.480 0.480 0.481 0.481 0.482
2.1 0.482 0.483 0.483 0.483 0.484 0.484 0.485 0.485 0.485 0.486
2.2 0.486 0.486 0.487 0.487 0.487 0.488 0.488 0.488 0.489 0.489
2.3 0.489 0.490 0.490 0.490 0.490 0.491 0.491 0.491 0.491 0.492
2.4 0.492 0.492 0.492 0.492 0.493 0.493 0.493 0.493 0.493 0.494
2.5 0.494 0.494 0.494 0.494 0.494 0.495 0.495 0.495 0.495 0.495
2.6 0.495 0.495 0.496 0.496 0.496 0.496 0.496 0.496 0.496 0.496

The above table shows the standard normal probability from 0 to z as seen at the left below.  The Excel functions use left to z as shown at the right below.

Standard normal distribution 0 to z: Table valuesStandard normal cumulative distribution left to z: Excel functions

Level of Confidence c Critical value zc
.80 1.28
.85 1.44
.90 1.645
.95 1.96
.99 2.58

Student's t Distribution.  T-values generated by Excel.

c 0.9 0.95 0.99 c 0.9 0.95 0.99
one tail 0.05 0.025 0.005 one tail 0.05 0.025 0.005
d.f. / two tail 0.1 0.05 0.01 d.f. / two tail 0.1 0.05 0.01
1 6.31 12.71 63.66 19 1.73 2.09 2.86
2 2.92 4.30 9.92 20 1.72 2.09 2.85
3 2.35 3.18 5.84 21 1.72 2.08 2.83
4 2.13 2.78 4.60 22 1.72 2.07 2.82
5 2.02 2.57 4.03 23 1.71 2.07 2.81
6 1.94 2.45 3.71 24 1.71 2.06 2.80
7 1.89 2.36 3.50 25 1.71 2.06 2.79
8 1.86 2.31 3.36 26 1.71 2.06 2.78
9 1.83 2.26 3.25 27 1.70 2.05 2.77
10 1.81 2.23 3.17 28 1.70 2.05 2.76
11 1.80 2.20 3.11 29 1.70 2.05 2.76
12 1.78 2.18 3.05 30 1.70 2.04 2.75
13 1.77 2.16 3.01 35 1.69 2.03 2.72
14 1.76 2.14 2.98 40 1.68 2.02 2.70
15 1.75 2.13 2.95 45 1.68 2.01 2.69
16 1.75 2.12 2.92 50 1.68 2.01 2.68
17 1.74 2.11 2.90 INF 1.64 1.96 2.58
18 1.73 2.10 2.88

Infant mortality data is from http://www.odci.gov/cia/publications/factbook/fields/infant_mortality_rate.html and from http://www.overpopulation.com/267
Income data is from http://www.un.org/Depts/unsd/social/inc-eco.htm

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