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 () is shown by the grey solid histogram columns. The first two questions that follow the graphs pertain to these four graphs.
Statistic  Equations  Excel 

Calculate a z value from an x  ^{z = }  =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 value given m and s  =STANDARDIZE(x, m, s/SQRT(n))  
Calculate an error tolerance E  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  
Calculate a confidence interval for a mean m for large n using the sample deviation s  
Calculate a confidence interval for a mean m for small n using the sample deviation s  
Calculate a critical tvalue for a two tailed tdistribution  =TINV(level of significance,degrees of freedom)


Calculate a critical tvalue for a one tailed tdistribution  =TINV(2*level of significance,degrees of freedom) 
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.
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.
Level of Confidence c  Critical value z_{c} 

.80  1.28 
.85  1.44 
.90  1.645 
.95  1.96 
.99  2.58 
Student's t Distribution. Tvalues 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 