It has been reported that 26% of children in the U.S. live in homes where there is a single parent. A school administrator cliamed that less than 26% of children in her school district lived in homes with a single parent. In a sample of 225 children, 54 of them lived in a home with a single parent. (a) Test the administrator's claim using a 5% level of significance using the classical approach and p-value approach. (b) construct a 90% confidence interval for the population proporation p of children living in home with a single parent. Does your result agree with your conclusion in part (a)?

To test the administrator's claim and calculate the confidence interval, we need to follow these steps:

Step 1: Define hypotheses.
Let's denote:
- p: The proportion of children in the school district who live in homes with a single parent.
- p0: The proportion reported for children in the entire U.S. (p0 = 0.26, as given)

The null hypothesis (H0) states that the proportion in the school district is equal to the national proportion: H0: p = p0
The alternative hypothesis (Ha) states that the proportion in the school district is less than the national proportion: Ha: p < p0

Step 2: Choose a significance level.
The significance level (α) is given as 5% (or 0.05).

Step 3: Calculate the test statistic.
For the classical approach, we use the z-test statistic. The formula for the z-test statistic is:
z = (p̂ - p0) / sqrt((p0 * (1 - p0)) / n)
where p̂ is the proportion in the sample (54/225), p0 is the proportion reported for the U.S. (0.26), and n is the sample size (225).

Step 4: Determine the critical value.
The critical value is found using the significance level (α) and the z-table. Since Ha: p < p0, we are looking for the critical value associated with the left tail of the distribution.

Step 5: Make a decision.
If the test statistic is less than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 6: Calculate the p-value.
The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. To calculate the p-value, we need to find the area under the standard normal curve to the left of the test statistic.

Once we have completed the hypothesis test, we can move on to constructing the confidence interval.

For the confidence interval, we will use the formula:
CI = p̂ ± z * sqrt((p̂ * (1 - p̂)) / n)
where p̂ is the proportion in the sample, z is the critical value associated with the desired confidence level (90%), and n is the sample size.

Now, let's begin with the calculations.

To test the school administrator's claim and construct a confidence interval, we can use hypothesis testing. Here's how you can approach this problem:

(a) Test the administrator's claim using the classical approach and p-value approach:

Step 1: Formulate the hypotheses:
- Null hypothesis (H0): The proportion of children living in homes with a single parent in the school district is equal to or greater than 26%. (p >= 0.26)
- Alternative hypothesis (Ha): The proportion of children living in homes with a single parent in the school district is less than 26%. (p < 0.26)

Step 2: Select the level of significance:
The problem states a 5% level of significance, so we will use alpha = 0.05.

Step 3: Calculate the test statistic (z-score):
The test statistic formula for testing proportions is:
z = (p̂ - p0) / √(p0 * (1 - p0) / n)

In this case:
p̂ = 54/225 = 0.24 (proportion from the sample)
p0 = 0.26 (proportion from the claim)
n = 225 (sample size)

z = (0.24 - 0.26) / √(0.26 * (1 - 0.26) / 225)

Calculating this will give you the test statistic.

Step 4: Determine the critical value or p-value:
For the classical approach, we will compare the test statistic to the critical value obtained from the standard normal distribution table. Since the alternative hypothesis states "less than," we are performing a one-tailed test. With alpha = 0.05, the critical z-score is approximately -1.645 (look up the value in the table).

For the p-value approach, we calculate the p-value corresponding to the test statistic by finding the area under the curve to the left of the test statistic. We then compare the p-value to the significance level (alpha) to make a decision.

Step 5: Make a decision and interpret the result:
- Classical approach: If the test statistic is less than the critical value (-1.645), we reject the null hypothesis in favor of the alternative hypothesis. If it is greater than or equal to the critical value, we fail to reject the null hypothesis.
- p-value approach: If the p-value is less than the significance level (alpha = 0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

(b) Construct a 90% confidence interval for the population proportion p:

To construct a confidence interval, we can use the following formula:
Confidence Interval = p̂ ± z * √(p̂ * (1 - p̂) / n)

Where:
p̂ = proportion from the sample
z = z-score corresponding to the desired confidence level (90% in this case)
n = sample size

By calculating this, we can find the lower and upper bounds of the confidence interval.

Now, you can apply these steps to find the solutions to parts (a) and (b) of the problem.