Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of $80$ dollars. New cost-cutting measures were started and a sample of $25$ claims was tested. The sample mean of the costs to process these claims was $76\%$ and the sample standard deviation of the costs was $\$10$. We would like to test whether the cost-cutting measures seem to be working at the $5\%$ significance level.
State the null and alternative hypotheses for this test. Here is the $t$-distribution table
$$\begin{array}{r|cccccccccccc}
\text{cum. prob} & t_{.60} & t_{.76} & t_{.80} & t_{.85} & t_{.90} & t_{.95} & t_{.975} & t_{.99} & t_{.995} & t_{.999} & t_{.9995} \\
\text{one-tail} & 0.50 & 0.25 & 0.20 & 0.15 & 0.10 & 0.05 & 0.025 & 0.01 & 0.005 & 0.001 & 0.0005 \\
\text{two-tails} & 1.00 & 0.50 & 0.40 & 0.30 & 0.20 & 0.10 & 0.05 & 0.02 & 0.01 & 0.002 & 0.001\\ \hline
\text{df} & & & & & & & & & & &\\
1 & 0.000 & 1.000 & 1.376 & 1.963 & 3.078 & 6.314 & 12.71 & 31.82 & 63.66 & 318.31 & 636.62\\
2 & 0.000 & 0.816 & 1.061 & 1.386 & 1.886 & 2.920 & 4.303 & 6.965 & 9.925 & 22.327 & 31.559\\
3 & 0.000 & 0.765 & 0.978 & 1.250 & 1.638 & 2.353 & 3.182 & 4.541 & 5.841 & 10.215 & 12.924 \\
4 & 0.000 & 0.741 & 0.941 & 1.190 & 1.533 & 2.132 & 2.776 & 3.747 & 4.604 & 7.173 & 8.610\\
5 & 0.000 & 0.727 & 0.920 & 1.156 & 1.476 & 2.015 & 2.571 & 3.365 & 4.032 & 5.893 & 6.869\\\hline
\color{blue}{6} & 0.000 & 0.718 & 0.906 & 1.134 & 1.440 & 1.943 & 2.447 & 3.143 & 3.707 & 5.208 & 5.959\\
\color{blue}{7} & 0.000 & 0.711 & 0.896 & 1.119 & 1.415 & 1.895 & 2.365 & 2.998 & 3.499 & 4.785 & 5.408\\
\color{blue}{8} & 0.000 & 0.706 & 0.889 & 1.108 & 1.397 & 1.860 & 2.306 & 2.896 & 3.355 & 4.501 & 5.041\\
\color{blue}{9} & 0.000 & 0.703 & 0.883 & 1.100 & 1.383 & 1.833 & 2.262 & 2.821 & 3.250 & 4.297 & 4.781\\
\color{blue}{10} & 0.000 & 0.700 & 0.879 & 1.093 & 1.372 & 1.812 & 2.228 & 2.764 & 3.169 & 4.144 & 4.587\\\hline
11 & 0.000 & 0.609 & 0.876 & 1.088 & 1.363 & 1.796 & 2.201 & 2.718 & 3.106 & 4.025 & 4.437\\
12 & 0.000 & 0.695 & 0.873 & 1.083 & 1.356 & 1.782 & 2.179 & 2.681 & 3.055 & 3.930 & 4.318\\
13 & 0.000 & 0.694 & 0.870 & 1.079 & 1.350 & 1.771 & 2.160 & 2.650 & 3.012 & 3.852 & 4.221\\
14 & 0.000 & 0.692 & 0.868 & 1.076 & 1.345 & 1.761 & 2.145 & 2.624 & 2.977 & 3.787 & 4.140\\
15 & 0.000 & 0.691 & 0.866 & 1.074 & 1.341 & 1.753 & 2.131 & 2.602 & 2.947 & 3.733 & 4.073\\\hline
\color{blue}{16} & 0.000 & 0.690 & 0.865 & 1.071 & 1.337 & 1.746 & 2.120 & 2.583 & 2.921 & 3.686 & 4.015\\
\color{blue}{17} &0.000 & 0.689 & 0.863 & 1.069 & 1.333 & 1.740 & 2.110 & 2.567 & 2.898 & 3.646 & 3.965\\
\color{blue}{18} & 0.000 & 0.688 & 0.862 & 1.067 & 1.330 & 1.734 & 2.101 & 2.552 & 2.878 & 3.610 & 3.922\\
\color{blue}{19} &0.000 & 0.688 & 0.861 & 1.066 & 1.328 & 1.729 & 2.093 & 2.539 & 2.861 & 3.579 & 3.883\\
\color{blue}{20} & 0.000 & 0.687 & 0.860 & 1.064 & 1.325 & 1.725 & 2.086 & 2.528 & 2.845 & 3.552 & 3.850\\\hline
21 & 0.000 & 0.686 & 0.859 & 10.63 & 1.323 & 1.721 & 2.080 & 2.518 & 2.831 & 3.527 & 3.819\\
22 & 0.000 & 0.686 & 0.858 & 1.061 & 1.321 & 1.717 & 2.074 & 2.508 & 2.819 & 3.505 & 3.792\\
23 & 0.000 & 0.685 & 0.858 & 1.060 & 1.319 & 1.714 & 2.069 & 2.500 & 2.807 & 3.485 & 3.768\\
24 & 0.000 & 0.685 & 0.857 & 10.59 & 1.318 & 1.711 & 2.604 & 2.492 & 2.797 & 3.467 & 3.745\\
25 & 0.000 & 0.684 & 0.856 & 1.058 & 1.316 & 1.708 & 2.060 & 2.485 & 2.787 & 3.450 & 3.7325\\\hline
\color{blue}{26} & 0.000 & 0.684 & 0.856 & 1.058 & 1.315 & 1.706 & 2.056 & 2.479 & 2.779 & 3.435 & 3.707\\
\color{blue}{27} & 0.000 & 0.684 & 0.855 & 1.057 & 1.314 & 1.703 & 2.052 & 2.473 & 2.771 & 3.421 & 3.690\\
\color{blue}{28} & 0.000 & 0.683 & 0.855 & 1.056 & 1.313 & 1.701 & 2.048 & 2.467 & 2.763 & 3.408 & 3.674\\
\color{blue}{29} & 0.000 & 0.683 & 0.853 & 1.055 & 1.311 & 1.699 & 2.045 & 2.462 & 2.756 & 3.396 & 3.659\\
\color{blue}{30} & 0.000 & 0.683 & 0.854 & 1.055 & 1.310 & 1.697 & 2.042 & 2.457 & 2.750 & 3.385 & 3.646\\\hline
40 & 0.000 & 0.681 & 0.851 & 1.050 & 1.303 & 1.684 & 2.021 & 2.423 & 2.704 & 3.307 & 3.551\\
60 & 0.000 & 0.679 & 0.848 & 1.045 & 1.296 & 1.671 & 2.000 & 2.380 & 2.660 & 3.232 & 3.460\\
80 & 0.000 & 0.677 & 0.845 & 1.042 & 1.290 & 1.660 & 1.984 & 2.364 & 2.626 & 3.174 & 3.390\\
100 & 0.000 & 0.675 & 0.842 & 1.037 & 1.282 & 1.646 & 1.962 & 2.330 & 2.581 & 3.098 & 3.300\\
1000 & 0.000 & 0.674 & 0.842 & 1.036 & 1.282 & 1.645 & 1.960 & 2.326 & 2.576 & 3.090 & 3.291\\\hline
\color{blue}{\text{z}} & 0.000 & 0.674 & 0.842 & 1.036 & 1.282 & 1.645 & 1.960 & 2.326 & 2.576 & 3.090 & 3.291\\\hline
\text{Confidence Level} & 0\% & 50\% & 60\% & 70\% & 80\% & 90\% & 95\% & 98\% & 99\% & 99.8\% & 99.9\%\\
\end{array}$$
Best Answer
I think instead of saying "the average cost to process a claim is approximately normally distributed with a mean of 80 dollars" you should have said "the cost to process a claim is approximately normally distributed with a mean of 80 dollars", letting the word "mean" be the only reference to averages.
The null hypothesis would be that the cost-cutting measures had no effect, so that that mean is still at least 80 dollars. The alternative hypothesis were effective, so that the mean is now lower than that.
Generally a null hypothesis is something presumed true until statistical evidence indicates otherwise.