The owner of a small pet supply store wants to open a second store in another city, but he only wants to do so if more than one-third of the city's households have pets (otherwise there won't be enough business). He samples 150 of the households and finds that 64 have pets.
Define the appropriate parameter(s) and state the hypotheses for testing if this sample provides evidence that more than one-third of households in this city own pets.

[tex]H_0 : p = 0.3333\\\\H_a : p > 0.3333\\\\n = 150\\\\x = 64\\\\\hat{p} = \frac{x}{n} = \frac{64}{150} = 0.4267\\\\P_0 = 0.3333\\\\1 - P_0 = 1 - 0.3333 = 0.6667\\\\[/tex]

[tex]z = \frac{\hat{p} - P_0}{[\sqrt{\frac{P_0 \times (1 - P_0 )}{n}}]}[/tex]

[tex]= \frac{0.4267 - 0.3333}{[\sqrt{\frac{(0.3333 \times 0.6667)}{150}}]}\\\\= 2.426[/tex]

Calculating the right-tailed test:

[tex]P(z > 2.426) = 1 - P(z < 2.426) = 1 - 0.9924 = 0.0076\\\\P-value = 0.0076\\\\\alpha = 0.05\\\\P-value < \alpha[/tex]

Therefore, we reject the null hypothesis, Its example shows that more than a third of the families own pets in this town.

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