A manager at ACME Equipment Sale and Rental wondered how offering a free two-year service...

Step 1
a)
All necessary assumptions are met:
The sampling method for each population is simple random sampling.
Independent samples.
Each sample includes at least 10 successes and 10 failures.
Proportion of customers purchasing tractors with warranties ${\displaystyle =p_1=\frac{93}{250}=0.372}$
Proportion of customers purchasing tractors without warranties ${\displaystyle =p_2=\frac{54}{250}=0.216}$
Difference in proportions ${\displaystyle =0.372-0.216=0.156}$
Pooled sample proportion ${\displaystyle =p=\frac{(p1\cdot n1+p2\cdot n2)}{(n1+n2)}=\frac{(0.372\cdot 250+0.216\cdot 250)}{500}=0.294}$
The standard error (SE) of the sampling distribution difference between two proportions.
${\displaystyle SE=\sqrt{p\times (1-p)\times (\frac{1}{n_1}+\frac{1}{n_2})}}$
where p is the pooled sample proportion, ${\displaystyle n_1}$ is the size of sample 1, and ${\displaystyle n_2}$ is the size of sample 2.
${\displaystyle SE=\sqrt{\{0.294\cdot (1-0.294)\cdot [\left(1\text{/}250\right)+\left(1\text{/}250\right)]\}}=0.04}$
Step 2
z value for $95\mathrm{\%}$ confidence interval is 1.96
Margin of error ${\displaystyle =z\cdot SE=1.96\cdot 0.04=0.0784}$
$95\mathrm{\%}$ confidence interval for the difference between the proportions
${\displaystyle =(0.156-0.0784,0.156+0.0784)}$
${\displaystyle =(0.077,0.235)}$
For $95\mathrm{\%}$ of the samples, the difference in proportions lies between 0.077 and 0.235
b)
Formulating Hypothesis
Null :${\displaystyle H_0}$: Offering the warranty does not increases the proportion of customers who eventually purchase a tractor.
Alternate : Ha: Offering the warranty increases the proportion of customers who eventually purchase a tractor.
Calculating Test statistic z :
${\displaystyle Z=\frac{\text{Difference in proportions}}{SE}=\frac{0.156}{0.04}}$
${\displaystyle Z=3.9}$
Decision about the null hypothesis
At 0.05 significance level
Since it is observed that ${\displaystyle \mid z\mid =3.9>Z_c=1.96}$, it is then concluded that the null hypothesis is rejected.
Using the P-value approach: The p-value is ${\displaystyle p=0.00004}$, and since P-vale ${\displaystyle <0.05}$, it is concluded that the null hypothesis is rejected.
Conclusion :
Offering the warranty increases the proportion of customers who eventually purchase a tractor.
Yes, they should offer a guarantee