Peter owns two bakeries. One of them is in a large city and the other is in a small town. He wants to see if more people prefer his triple chocolate cake over his chocolate cream pie in the town than in the city.
Peter has compiled a list of $3{,}000$ regular customers that visit his city bakery and $1{,}000$ that visit his town bakery. He randomly selects $200$ customers from the city location and $50$ from the town location and discovers that $120$ of the city customers sampled and $45$ of the town customers sampled prefer the triple chocolate cake over the chocolate cream pie. Peter then asks his statistician friend for help constructing a $95\%$ confidence interval for the difference between the proportion of people who prefer the triple chocolate cake over the chocolate cream pie at the town bakery and at the city bakery.
**The friend tells him:**
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