Find the first seven terms of the Maclaurin expansion of $f(z)=\sin ^{2}(z)$ in four ways, as follows. (a) First, compute the Taylor coefficients at 0. (b) Find the first seven terms of the product of the Maclaurin series for sin(z) with itself. (c) Write $\sin ^{2}(z)$ in terms of the exponential function and use the Maclaurin expansion of this function. (d) Write $\sin ^{2}(z)=(1-\cos (2 z)) / 2$ and use the Maclaurin expansion of cos(z).

In his book Outliers, author Malcolm Gladwell argues that more American-born baseba]] players have birth dates in the months immediately following July 31 because that was the age cutoff date for nonschool baseball leagues. The table below lists month of births for a sample of American-born basebal l players and foreign-born baseball players. Using a 0.05 significance level, is there sufficient evidence to warrant rejection of the claim that months of births of baseball players are independent of whether they are born in America? Do the data appear to support Gladwell 's claim?

$$\begin{matrix} \text{ } & \text{Jan. } & \text{Feb.} & \text{March} & \text{April} & \text{May} & \text{June} & \text{July} & \text{Aug.} & \text{Sep.} & \text{Oct.} & \text{Nov.} & \text{Dec.}\\ \text{Born in America} & \text{387} & \text{329} & \text{366} & \text{344} & \text{336} & \text{313} & \text{313} & \text{503} & \text{421} & \text{434} & \text{398} & \text{371}\\ \text{Foreign Born} & \text{101} & \text{82} & \text{85} & \text{82} & \text{94} & \text{83} & \text{59} & \text{91} & \text{70} & \text{100} & \text{103} & \text{82}\\ \end{matrix}$$

Members of a certain club are required to register for one of three games: badminton, table tennis or darts. The number of club members of each gender choosing each game in a particular year is shown in the table.

$$\begin{matrix} \text{ } & \text{Badminton} & \text{Table tennis} & \text{Darts}\\ \text{Male} & \text{37} & \text{16} & \text{28}\\ \text{Female} & \text{32} & \text{10} & \text{19}\\ \end{matrix}$$

Use a chi-squared test, at the 5% significance level, to test whether choice of game is independent of gender. State clearly the null and alternative hypotheses, the expected values and the number of degrees of freedom.