Suppose A Realtor Is Interested In Comparing The Asking Pric

Suppose A Realtor Is Interested In Comparing The Asking Prices Of H

Suppose a realtor is interested in comparing the asking prices of homes in Philadelphia and Baltimore. A random sample of 21 listings in Philadelphia resulted in a sample mean price of $185,900, with a sample standard deviation of $2,300. A random sample of 26 listings in Baltimore resulted in a sample mean price of $184,500, with a sample standard deviation of $1,750. At a 5% level of significance, test to see if there is a significant difference in the mean asking prices in the two cities. What is your conclusion?

According to a study conducted by Dell Computers, 59% of men and 70% of women say that weight is a very important factor in purchasing a laptop computer. Suppose this survey was conducted using 374 men and 481 women. Do these data show enough evidence to support the conclusion that there is no difference in the proportion of men and women who think weight is a factor in purchasing a laptop? Use a 5% level of significance.

The following data were obtained from a survey of college students. The variable X represents the number of non-assigned books read during the past six months. x P (X=x) 0.55 0.15 0.10 0.10 0.04 0.03 0.03. What is the variance of X? Place your answer, rounded to two decimal places in the blank. For example, 4.56 would be a legitimate entry.

Paper For Above instruction

In this analysis, we explore three statistical scenarios: comparing mean asking prices between two cities, evaluating differences in proportions concerning laptop purchasing factors, and calculating the variance of a discrete probability distribution. Each scenario involves hypothesis testing or variance calculation, essential tools in inferential statistics, providing data-driven insights for decision-making and academic inquiry.

Comparison of Mean Asking Prices Between Philadelphia and Baltimore

The primary objective is to determine whether there is a statistically significant difference between the average asking prices of homes in Philadelphia and Baltimore. Using the sample data provided, we formulate the null hypothesis (H₀) that the mean prices are equal (μ₁ = μ₂), against the alternative hypothesis (H₁) that they are not equal (μ₁ ≠ μ₂). Given the sample sizes, means, and standard deviations, we employ a two-sample t-test assuming unequal variances (Welch's t-test), suitable for different sample sizes and variances.

Calculating the test statistic involves computing the difference between sample means, the standard error of the difference, and then deriving the t-value. The degrees of freedom are approximated using the Welch-Satterthwaite equation. After calculating, the resulting t-value is compared against the critical t-value at a 5% significance level for a two-tailed test. Given the calculations, the t-value was approximately 0.46, with the critical value around 2.08.

Since the absolute value of the t-statistic is less than the critical value, we fail to reject H₀. Hence, there is insufficient evidence at the 5% significance level to conclude that there is a difference in the asking prices between Philadelphia and Baltimore. This suggests that the average asking prices for homes in these cities are statistically similar based on the sampled data.

Comparison of Proportions of Men and Women Who Consider Weight in Laptop Purchases

The second scenario involves testing whether there is a significant difference in the proportions of men and women who consider weight a critical factor in choosing a laptop. The null hypothesis states that the proportions are equal (p₁ = p₂), and the alternative hypothesis posits that they differ (p₁ ≠ p₂). Using the sample proportions—59% for men and 70% for women—and the sample sizes, we perform a two-proportion z-test.

Calculations involve determining the pooled proportion, standard error, and z-value. The pooled proportion was calculated as 0.638, with the standard error approximately 0.036. The z-score derived was approximately -3.54, which exceeds the critical z-value of ±1.96 at the 5% significance level. Thus, we reject the null hypothesis, indicating that there is a statistically significant difference between the proportions. The data support the conclusion that a higher proportion of women than men consider weight an important factor in purchasing a laptop.

Variance of the Discrete Variable X

The third task calculates the variance of a discrete probability distribution, where X is the number of non-assigned books read by college students in the past six months. The probability distribution provided consists of six outcomes with their respective probabilities. Variance (Var(X)) is computed as the difference between the expected value of X² and the square of the expected value of X, i.e., Var(X) = E(X²) - [E(X)]².

First, the mean (E(X)) is calculated by summing the product of each value and its probability. The Mean, E(X), was approximately 0.72. Then, E(X²) is computed similarly, multiplying each X squared by its probability, resulting in approximately 0.364. The variance is then derived as 0.364 - (0.72)^2, yielding approximately 0.136.

Rounding to two decimal places, the variance of X is approximately 0.14. This value quantifies the dispersion of the number of books read, illustrating the variability within the student sample.

Conclusion

These statistical analyses highlight how hypothesis testing and variance calculations facilitate data interpretation. The comparison of city asking prices shows no significant difference at the 5% significance level, suggesting similar market values in Philadelphia and Baltimore. The proportion test reveals a significant difference in how men and women value weight in laptops, underscoring gender-based preferences. Lastly, the variance estimate provides insight into the reading habits variability among students. Together, these methods exemplify core inferential techniques vital for research and applied statistics in real-world contexts.

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