MS 150 Statistics Fall 2000 T2

On the following four graphs the distribution of the x data is shown by the black outline histogram columns.  The x distribution is the same for all four graphs.   The population mean for the x distribution is m = 8.  The distribution of 40 sample means (xbar.gif (842 bytes)) is shown by the grey solid histogram columns.  The first two questions that follow the graphs pertain to these four graphs.

t2.gif (9698 bytes)

  1. What is the shape of the x distribution?


  2. Choose the letter for the graph which shows a likely distribution of the forty sample means (xbar.gif (842 bytes) distribution).
Statistic Equations Excel
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 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 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 =TINV(level of significance,degrees of freedom)

 
Calculate a critical t-value for a one tailed t-distribution =TINV(2*level of significance,degrees of freedom)

normal_curve.jpg (22909 bytes)

During the past year I've run from the College to my house in Nett in a mean time of m = 91 minutes with a standard deviation of s = 6.5 minutes.  My arrival times are normally distributed about the mean.    Use this data for the following problems.

  1. What percentage of runs were completed between 84.5 and 97.5 minutes?


  2. I telephone home before I leave the College so that my wife will know my departure time.   She wants to know the expected time x in minutes by which there is a 90% probability of my arrival.  [Hint: Note that 10% of the arrivals will be at a time beyond the expected time x].


  3. The above mean time is based on a sample of 60 runs over the past year.  I want to predict my mean time for the coming year.  What would be the point estimate value for next year's mean time?


  4. Find a 95% confidence interval for next year's mean time using m = 91 minutes , s = 6.5 minutes, and n = 60 runs.


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

 

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