A city government claims that no more than $25\%$ of households have solar panels. A researcher suspects the rate is actually higher and surveys $200$ households, finding that $62$ have solar panels. Test if there is evidence that more than $25\%$ of households have solar panels using $\alpha =0.01$.

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:

$\alpha =0.01$, $n=40$, $x=28$, & claim is $p=0.75$

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:

$\alpha =0.10$, $n=100$, $x=42$, & claim is $p=0.25$

A snack company claims that at least 70% of people prefer its new low-sugar granola bar over the original version. To test this claim, a grocery chain surveys a random sample of 80 customers, & 50 say they prefer the new version. Use $\alpha =0.10$ to test whether more than 70% of customers prefer the new granola bar. Should the grocery chain stock more of the new product & reduce shelf space for the original version?

A cereal company advertises that $50\%$ of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey $200$ households, and $114$ report that they purchase the brand at least once per month.

(A) Use $\alpha =0.10$ to test the claim.

A cereal company advertises that $50\%$ of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey $200$ households, and $114$ report that they purchase the brand at least once per month.

$\left(B\right)$ Based on the research, should the company change its advertising?