The Speed Of Automobiles On A Section Of I 95 Is Normally Distribute
The speed of automobiles on a section of I-95 is normally distributed with a population mean of 65 miles per hour and a population standard deviation of 8 miles per hour. A random sample of 30 cars is to be selected for a speed study. Several statistical questions are posed based on this data, including the properties of the sampling distribution of the sample mean and the probabilities related to different speed intervals.
Additionally, the assignment involves analyzing sample data regarding automobile speeds, comparing asking prices in two cities, and examining differences in proportions from survey data concerning laptop purchasing factors. These diverse statistical scenarios require understanding sampling distributions, hypothesis testing, and confidence intervals, all within the context of real-world applications.
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The study of automobile speeds on highways provides valuable insights into traffic flow and safety management. Given that the speeds on a section of I-95 are normally distributed with a mean of 65 miles per hour and a standard deviation of 8 miles per hour, we can examine the sampling distribution of the sample mean and calculate probabilities associated with various speed thresholds.
First, considering the sampling distribution of the mean for a sample size of 30 cars, the shape of this distribution is approximately normal due to the central limit theorem, which states that the distribution of the sample mean will be normal if the sample size is sufficiently large, regardless of the population distribution. Since the population distribution is normal and the sample size of 30 is generally considered sufficient, the sampling distribution will also be normal with a mean equal to the population mean of 65 miles per hour and a standard deviation (standard error) calculated as σ/√n, where σ is 8 miles per hour and n is 30. This yields a standard error of 8/√30 ≈ 1.46 miles per hour.
Next, calculating the probability that the sample mean will be 69 miles per hour or more involves computing the z-score for 69 miles per hour: (69 - 65)/1.46 ≈ 2.74. Consulting standard normal distribution tables, the probability that the z-score exceeds 2.74 is approximately 0.0031. This indicates that there is about a 0.31% chance that the average speed of the sample exceeds 69 miles per hour.
Similarly, to find the probability that the sample mean falls between 62 and 68 miles per hour, we compute their respective z-scores: (62 - 65)/1.46 ≈ -2.05 and (68 - 65)/1.46 ≈ 2.05. Looking these up in the standard
normal table, the probabilities correspond to about 0.0197 for z = -2.05 and 0.9803 for z = 2.05. Therefore, the probability that the sample mean is between 62 and 68 miles per hour is approximately 0.9803 - 0.0197 ≈ 0.9606, or 96.06%. If the sample size were increased to 60, the standard error would halve, as it is proportional to 1/√n. The new standard error becomes 8/√60 ≈ 1.03 miles per hour. Recalculating the z-scores and probabilities with this smaller standard error would yield even narrower confidence intervals and higher probabilities for the same speed ranges, reflecting increased precision with larger samples.
Moreover, the probability that the sample mean is less than 63 miles per hour involves computing the z-score: (63 - 65)/1.46 ≈ -1.37. The probability associated with z = -1.37 is approximately 0.0853, indicating an 8.53% chance that the average speed falls below 63 miles per hour.
Moving to the second scenario, where the population proportion of adults with a college degree is 0.35, a sample of 500 adults is used to estimate this proportion. The sampling distribution of the sample proportion can be modeled as approximately normal, provided np and n(1 - p) are both greater than 5. Calculating the standard deviation of this distribution, known as the standard error, involves the formula √[p(1 - p)/n], which results in √[0.35 * 0.65 / 500] ≈ 0.0209. The mean of this distribution is the population proportion, 0.35.
To determine the probability that the observed sample proportion falls within 0.03 of the true proportion, we look at the interval from 0.32 to 0.38. Computing the corresponding z-scores: (0.32 - 0.35)/0.0209 ≈ -1.43 and (0.38 - 0.35)/0.0209 ≈ 1.43, the associated probabilities are approximately 0.0764 for z = -1.43 and 0.9236 for z = 1.43. Thus, the probability that the sample proportion is within 0.03 of 0.35 is roughly 0.9236 - 0.0764 = 0.8472, or 84.72%.
In the third analysis, comparing home asking prices in Philadelphia and Baltimore involves hypothesis testing. The sample from Philadelphia has a mean of $185,900 with a standard deviation of $2,300 over 21 homes; Baltimore's mean is $184,500 with an SD of $1,750 over 26 homes. To test if the mean prices differ significantly at the 5% significance level, we perform a two-sample t-test, considering the unequal variances using the Welch's t-test formula.
The t-statistic is calculated as (185,900 - 184,500) / √[(2,300)^2/21 + (1,750)^2/26] ≈ 1,400 / √[(529,000/21) + (3,062,500/26)] ≈ 1,400 / √[(25,190.48) + (117,788.46)] ≈ 1,400 / √142,978.94 ≈ 1,400 / 378.12 ≈ 3.70. The degrees of freedom are approximated using the Welch-Satterthwaite equation, leading to roughly 40 degrees of freedom.
Consulting t-distribution tables, a t-value of 3.70 with about 40 degrees of freedom yields a p-value below 0.001, which is less than the significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is a statistically significant difference in asking prices between the two cities, with Philadelphia's homes priced higher on average.
Lastly, analyzing data from a survey conducted by Dell Computers regarding the importance of weight in purchasing laptops involves hypothesis testing for the difference in two proportions. With 374 men and 481 women surveyed, 59% of men and 70% of women consider weight a very important factor. The null hypothesis states that there is no difference in the proportions (p1 = p2).
The pooled proportion is (0.59 * 374 + 0.70 * 481) / (374 + 481) ≈ (220.66 + 336.7) / 855 ≈ 0.707. The standard error for the difference in proportions is √[p(1 - p) * (1/n1 + 1/n2)] ≈ √[0.707 * 0.293 * (1/374 + 1/481)] ≈ 0.043.
Calculating the z-score: (0.59 - 0.70) / 0.043 ≈ -2.56. The p-value associated with z = -2.56 is approximately 0.0104, which is less than the significance level of 0.05. Therefore, there is sufficient evidence to reject the null hypothesis and conclude that there is a statistically significant difference in the proportions of men and women who consider weight an important factor in purchasing a laptop, with women more likely to do so.
In conclusion, these statistical analyses demonstrate the importance of understanding sampling distributions, hypothesis testing, and probability calculations in making informed decisions based on data. The scenario involving vehicle speeds showcases the application of the central limit theorem and normal distribution properties, while comparative analyses of home prices and consumer preferences highlight the utility of t-tests and proportion tests in real-world contexts. Mastery of these concepts enables more accurate interpretation of data and supports evidence-based decision-making in various fields, including transportation safety, real estate, and consumer behavior.
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