To answer this question, we'll first need to find out the sample proportion, standard deviation under the null hypothesis, and the test statistic Z.
Let's begin by calculating the sample proportion. We know that the sample size is 200 and the number of successful outcomes is 107. So, we calculate the sample proportion by dividing the number of successful outcomes by the sample size, which is 107/200.
Next, we'll calculate the standard deviation under the null hypothesis. This involves the population proportion which is given as p=0.53. The formula for the standard deviation under the null hypothesis is:
sqrt([p*(1-p)]/n)
By substituting the known values into this formula, we get the standard deviation as approximately 0.0353.
Now we are almost into finding the test statistic Z value. The formula to calculate Z is:
(sample proportion - population proportion)/standard deviation
Substitute the calculated standard deviation and known sample and population proportions into the formula, and we get Z value as approximately 0.142.
And that's your answer, which is closest to option a) 0.15. This is the test statistic against the null hypothesis that the true population proportion is 0.53. This indicates that, given the assumed population proportion of 0.53, the probability of finding a sample proportion as extreme as the observed is relatively high, hence, we would not reject the null hypothesis at the 0.01 level of significance.