To solve the question, first, we have to standardize the data, meaning, we will compute the z-scores that correspond to 43, 38, 37, and 42.
Z-scores:
[tex]\begin{gathered} Z_{43}=\frac{43-41.2}{6.1}\approx0.29508, \\ Z_{38}\approx-0.52459, \\ Z_{37}\approx=-0.68852, \\ Z_{42}\approx0.13115. \end{gathered}[/tex]Now, using tables, we get the area under the normal curve that corresponds to each z-score to get the percentages for questions a, and b:
a) x<43 corresponds to 0.61603, therefore, 61.60% is below 43 inches.
b) x>38 corresponds to 0.70007, therefore, 70.01% is above 38 inches.
Now, using tables, we get that:
x<37 corresponds to 0.24556, and x<42 corresponds to 0.55217, therefore
[tex]0.55217-0.24556[/tex]corresponds to the interval between 37 and 42 inches.
a) 61.60%,
b) 70.01%,
c) 30.67%.
