91Ó°ÊÓ

A random sample of 1500 residcntial tclephones in Phoenix in 1990 indicated that 387 of the numbers were unlisted. A random sample of 1200 telcphones in the same year in Scottsdale indicated that 310 were unlisted. (a) Find a \(95 \%\) confidence interval on the difference in the two proportions and use this confidence interval to determine 2010 whether there is a statistically significant difference in proportions of unlisted numbers between the two cities. (b) Find a 90 \& confidence interval on the difference in the two proportions and use this confidence interval to determine if there is a statistically significant difference in proportions of unlisted numbers for the two cities. (c) Suppose that all the numbers in the problem description were doubled. That is, 774 residents of 3000 sampled in Phoenix and 620 residents of 2400 in Scottsdale had unlisted phone numbers. Repeat parts (a) and (b) and comment on the effect of increasing the sample size without changing the proportions on your results.

Step by step solution

01

Determine Sample Proportions

Calculate the sample proportions of unlisted numbers for each city. For Phoenix, the proportion \( p_1 = \frac{387}{1500} = 0.258 \). For Scottsdale, the proportion \( p_2 = \frac{310}{1200} = 0.2583 \).

02

Calculate the Standard Error for Difference in Proportions

The formula for the standard error of the difference between two proportions is \( SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} } \), where \( n_1 = 1500 \) and \( n_2 = 1200 \). Substituting the known values, calculate \( SE = \sqrt{ \frac{0.258 \times 0.742}{1500} + \frac{0.2583 \times 0.7417}{1200} } \approx 0.0232 \).

03

Find the 95% Confidence Interval

The critical value for a 95% confidence interval is \( z = 1.96 \). The confidence interval is \( p_1 - p_2 \pm z \times SE = 0.258 - 0.2583 \pm 1.96 \times 0.0232 \approx (-0.0241, 0.0235) \). The interval includes zero, indicating no statistically significant difference at the 95% confidence level.

04

Find the 90% Confidence Interval

The critical value for a 90% confidence interval is \( z = 1.645 \). Calculate the interval: \( 0.258 - 0.2583 \pm 1.645 \times 0.0232 \approx (-0.0201, 0.0195) \). This interval also includes zero, indicating no statistically significant difference at the 90% confidence level.

05

Double the Sample Sizes

For the new sample sizes, calculate proportions \( p_1 = \frac{774}{3000} = 0.258 \) and \( p_2 = \frac{620}{2400} = 0.2583 \). The standard error becomes \( SE = \sqrt{ \frac{0.258 \times 0.742}{3000} + \frac{0.2583 \times 0.7417}{2400} } \approx 0.0164 \).

06

Recalculate 95% Confidence Interval with Increased Sample Sizes

With doubled sample sizes, using \( z = 1.96 \), the confidence interval becomes \( 0.258 - 0.2583 \pm 1.96 \times 0.0164 \approx (-0.0175, 0.0169) \). Zero is included again, indicating no significant difference.

07

Recalculate 90% Confidence Interval with Increased Sample Sizes

For a 90% confidence level and doubled sample sizes, the interval is \( 0.258 - 0.2583 \pm 1.645 \times 0.0164 \approx (-0.0148, 0.0142) \). Again, includes zero, reaffirming no significant difference.

08

Analyze Effects of Sample Size Increase

Increasing the sample size reduced the standard error, thus narrowing the confidence intervals but did not change the result of no statistical significance, as zero is still within all intervals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportions

Sample proportions are a way of estimating the true proportion of a given population by considering just a sample of it. In statistics, these proportions are essential for understanding the distribution of certain features.
For example, in the given exercise, the proportion of unlisted phone numbers in Phoenix (\(p_1\)) is calculated as \(\frac{387}{1500} = 0.258\), while in Scottsdale (\(p_2\)), it's \(\frac{310}{1200} = 0.2583\). These proportions tell us the part of the sample in each city that has unlisted numbers.
The calculation of sample proportions is crucial for determining other statistical measures like confidence intervals, which rely heavily on these initial numbers.

Standard Error

The standard error measures the variability of a sample statistic, in this case, the difference between two sample proportions. It helps us understand the expected variation in the sample proportions if the survey were repeated multiple times.
The standard error for the difference in proportions is calculated using the formula: \[SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }\] Where \(n_1\) and \(n_2\) are the sizes of the samples, and \(p_1\) and \(p_2\) are the sample proportions.
In our exercise, it was calculated as approximately 0.0232 for the original sample sizes, indicating the extent to which the difference in the sample proportions could vary. Lower standard error implies higher precision in confidence intervals.

Statistical Significance

Statistical significance is a way to determine if the observed difference in sample proportions is not due to random chance. When analyzing confidence intervals, we look to see if zero is outside the interval to speak of statistical significance.
In our example, the 95% confidence interval was calculated as \((-0.0241, 0.0235)\), and for 90% as \((-0.0201, 0.0195)\). Since zero was included in both intervals, there is no statistically significant difference between the phone number listing patterns in Phoenix and Scottsdale. If zero were outside these intervals, we could claim statistical significance.

Sample Size Effect

Sample size plays a pivotal role in statistical analyses. Larger sample sizes generally lead to smaller standard errors, resulting in narrower confidence intervals and an increased precision of the estimate.
When the sample size in our problem was doubled, the standard error decreased from approximately 0.0232 to about 0.0164. Consequently, the confidence intervals also narrowed: \((-0.0175, 0.0169)\) for 95%, and \((-0.0148, 0.0142)\) for 90%.
However, even with a larger sample size, statistical significance was not observed as zero remained within the confidence intervals. Nonetheless, it is evident that a larger sample size could deepen our insights into the data's behavior by reducing variability.