Joe likes coin tricks. In one of his tricks, he flips a coin $3$ times. Let $X$ be the number of heads that appear. Joe will repeat this trick so that we have $n$ samples from $X$. Although Joe doesn't make any claim about whether the coin is fair, he does claim that he is flipping the coin all three times independently. We are skeptical that Joe might be deceiving you so you decide to test the claim.
(i) If we let our null hypotheses be Joe's claim, explain why we express it as $H_0: X$ is binomial $(3, p)$.
(ii) We take a random sample of size $n = 100$. For $i = 0, ..., 3$, let $Y_i$ be the number of samples of $X$ that result in $i$ many heads. We observe that $Y_0 = 43, Y_1 = 8, Y_2 = 5, Y_3 = 44$. What is the conclusion at significance level $\alpha = 0.10$?
What I did is as follows
(i) Not quite sure what to say here. Hypothesis testing begins with a claim about a population parameter; in this case, Joe claims that he flips the coin independently, and if he flips it $3$ times, then that is characteristic of a binomial distribution with some unknown probability of flipping the coin, $p$ (it is unknown since Joe hasn't made a claim about whether it is fair or not). Hence, our stated or null hypothesis is expressed that way.
(ii) We use a chi-squared goodness of fit test.
$H_0: p_0 = P(X = 0) = (1-p)^{3}, p_1 = P(X = 1) = 3p(1-p)^{2}, p_2 = P(X = 2) = 3p^{2}(1-p), p_3 = P(X = 3) = p^{3}$
The test statistic, $T$ is:
$$T = \frac{(43 - 100(1-p)^3)^2}{100(1-p)^3} + \frac{(8 - 100(3p)(1-p)^2)^2}{100(3p)(1-p)^2} + \frac{(5 - 100(3p^2)(1-p))^2}{100(3p^2)(1-p)} + \frac{(44 - 100p^3)^2}{100p^3}$$
From the tables, $\chi^2_{0.10}(3) = 6.251$
We reject the null hypothesis if $T > 6.251$.
If you plot $T$ against $p$, you get a somewhat parabolic shaped curve with minimum at $p = 0.503$ and $T = 205.119$. Regardless of what value of $p \in (0, 1)$ is chosen, $T$ will always be greater than $6.251$. So we reject the null hypothesis.
Is what I have done correct? Any assistance appreciated.
I would phrase (i) like this. Joe's claim is that the three flips are independent events. Also implied in the claim (but not explicitly stated) is that they are identically distributed; i.e., the probability of heads on any given flip does not change from one flip to another.
Consequently, the claim is equivalent to asserting that:
Therefore, to test whether there is sufficient evidence to suggest this claim is not true, the hypothesis to be tested is:
$$H_0 : X \sim \operatorname{Binomial}(3, p) \quad \text{vs.} \quad H_1 : X \not\sim \operatorname{Binomial}(3,p).$$
As for (ii), your approach and conclusion are correct, although when $p \approx 0.502723$, I get $T = 30312.9$.