Question

As of March, 'e' is going to be the same for Arizona? If so, then in any sample of Florida eligible residents, the proportion of fully vaccinated could be expected to be around 65%. Suppose you take a random sample of eligible Arizona residents and find that out of 1025 people, 615 are fully vaccinated. The sample proportion of eligible Arizona residents who are fully vaccinated is __% (round to the whole number percent). Use this sample to find a 95% confidence interval and conclude: I am 95% confident that the proportion of all eligible Arizona residents who are fully vaccinated is between __% and __% (round to whole number percent). Based on your interval, can you confidently claim that the proportion of all eligible Arizona residents who are fully vaccinated is less than the proportion of all fully vaccinated US adults, which is 65%? (Yes / No) Is the sample proportion significant enough that the proportion of Arizona residents who are fully vaccinated is less than the proportion of US adults? Test hypothesis at 5% significance level, null hypothesis would be H0: p = __ (enter decimal number). Alternate hypothesis would be Ha: p < __ (enter decimal number). The z-stat is __ (round to 4 decimal places), p-value is __ (round to 4 decimal places). The p-value is (less / more) than the level of significance. Conclusion: We (do / do not) have sufficient evidence to reject the null hypothesis. We (can / cannot) claim that the proportion of eligible Arizona residents who are fully vaccinated is less than the proportion of fully vaccinated US adults.

Answer 1

The sample proportion of eligible Arizona residents who are fully vaccinated is calculated to be 60%. A 95% confidence interval for the proportion of all eligible Arizona residents who are fully vaccinated is estimated to be between 57% and 63%. The hypothesis test shows that there is sufficient evidence to conclude that the proportion of eligible Arizona residents who are fully vaccinated is less than the proportion of fully vaccinated US adults.

Step-by-step explanation:

The sample proportion of eligible Arizona residents who are fully vaccinated can be calculated by dividing the number of fully vaccinated residents by the total number of residents in the sample:

Sample proportion = (Number of fully vaccinated residents) / (Total number of residents in the sample)

Sample proportion = 615 / 1025

Sample proportion = 0.6

To find a 95% confidence interval, we can use the formula:

Confidence interval = Sample proportion ± (Critical value) * (Standard error)

The critical value for a 95% confidence interval is 1.96 (from the standard normal distribution).

The standard error can be calculated using the formula:

Standard error = sqrt((Sample proportion * (1 - Sample proportion)) / Sample size)

For the given sample size of 1025, the standard error would be:

Standard error = sqrt((0.6 * (1 - 0.6)) / 1025)

Standard error ≈ 0.015

Substituting the values into the confidence interval formula:

Confidence interval = 0.6 ± 1.96 * 0.015

Confidence interval ≈ (0.570, 0.630)

Based on the confidence interval, we can conclude that with 95% confidence, the proportion of all eligible Arizona residents who are fully vaccinated is estimated to be between 57% and 63%.

To test the null hypothesis that the proportion of fully vaccinated Arizona residents is less than the proportion of fully vaccinated US adults (65%), we can use a hypothesis test.

The null hypothesis (H0) is that the proportion (p) of fully vaccinated Arizona residents is equal to or greater than 65%.

The alternative hypothesis (Ha) is that the proportion (p) of fully vaccinated Arizona residents is less than 65%.

The test statistic for this hypothesis test is the z-statistic, which can be calculated using the formula:

z-stat = (Sample proportion - Hypothesized proportion) / sqrt((Hypothesized proportion * (1 - Hypothesized proportion)) / Sample size)

Substituting the values into the formula:

z-stat = (0.6 - 0.65) / sqrt((0.65 * (1 - 0.65)) / 1025)

z-stat ≈ -5.501

The p-value can be calculated by finding the area under the standard normal distribution curve to the left of the z-statistic.

Using a standard normal table or calculator, the p-value for a z-statistic of -5.501 is less than 0.0001 (rounded to 4 decimal places).

The p-value is less than the level of significance (0.05), so we reject the null hypothesis.

We have sufficient evidence to conclude that the proportion of eligible Arizona residents who are fully vaccinated is less than the proportion of fully vaccinated US adults.