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?

• A. Yes, the claim is false.
• B. No, the claim is true.
• C. More info is required.

Answer: (A)

Verified step by step guidance

1. Identify the null hypothesis (H_0) and the alternative hypothesis (H_a). Here, H_0: p = 0.50 (the proportion of households purchasing the brand is 50%), and H_a: p \neq 0.50 (the proportion is different from 50%).
2. Calculate the sample proportion \( \hat{p} \) using the survey data: \( \hat{p} = \frac{114}{200} \).
3. Compute the standard error (SE) of the sample proportion under the null hypothesis using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] where \( p = 0.50 \) and \( n = 200 \).
4. Calculate the test statistic (z) using the formula: \[ z = \frac{\hat{p} - p}{SE} \]. This measures how many standard errors the sample proportion is away from the hypothesized proportion.
5. Determine the p-value corresponding to the calculated z-value for a two-tailed test. Compare the p-value to the significance level (commonly 0.05) to decide whether to reject H_0. If rejected, the company should consider changing its advertising; if not, no change is needed.