9.13 Hypothesis testing: two column model step by step example of

We have univariate, categorical data. Therefore, we can perform a one proportion z-test to test this belief. Our model will be
$N(p_o,\sqrt{\frac{p_o(1-p_o)}{n}})$
$=\; N(0.3,\sqrt{\frac{0.3(1-0.3)}{150}})$

1. Express the hypothesis to be tested in symbolic form,
2. Write a symbolic expression that must be true when the original claims is false.
3. The null hypothesis is the statement which includes the equality.
4. The alternative hypothesis is the statement without the equality.

Null Hypothesis in words: The null hypothesis is that the true population proportion of households that own at least three cell phones is equal to 30%.
Null Hypothesis symbolically:
$H_o$ : p = 30%
Alternative Hypothesis in words: The alternative hypothesis is that the population proportion of households that own at least three cell phones is more than 30%.
Alternative Hypothesis symbolically:
$H_o$ : p>30%

Randomization Condition: The problem tells us she used a random sample.
Independence Assumption: When you know you have a random sample it is likely outcomes are independent. There is no reason to think how many cell phones one household owns has any bearing on the next household.
10% Condition: I will assume that the city Michele lives in is large and that 150 households is less than 10% of all households in her community.
Success/Failure:
p ₀ (n)>10 and (1 - p ₀ )n>10
To meet this condition both the success and failure products must be larger than 10 (p ₀ is the value of the null hypothesis in decimal form.)
0.3(150) = 45>10 and (1 – 0.3)150 = 105>10

The conditions are satisfied, so we will use a hypothesis test for a single proportion to test the null hypothesis. For this calculation we need the sample proportion, $\hat{p}$
$\hat{p}=\frac{53}{150}=\; 0.3533$
$z=\frac{\hat{p}-p_o}{\sqrt{\frac{p_o(1\; -p_o)}{n}}}=\frac{0.3533\; -\; 0.3}{\sqrt{\frac{0.3(1\; -\; 0.3)}{150}}}=\frac{0.0533}{0.0374}=1.425$

Determine the P-value

State whether you reject or fail to reject the Null hypothesis.

If you reject the null hypothesis, continue to complete the following

Calculate and display your confidence interval for the Alternative hypothesis.

Conclusion State your conclusion based on your confidence interval.