Do American Coin Dealers Feel That Speculating In Silver Eag

58 Of American Coin Dealers Feel That Speculating In Silver Eagle

58% of American coin dealers feel that speculating in Silver Eagles will be profitable next year. In a simple random sample of 150 coin dealers, what is the probability that between 54% and 60% believe that it will be a good year to speculate on Silver Eagles? A. 0.9123 B. 0.1256 C. 0.3432 D. 0.5297 E. 0.

The average monthly mortgage payment for recent home buyers in Lexington is $732, with a standard deviation of $421. A random sample of 125 recent home buyers is selected. The approximate probability that their average monthly mortgage payment will be more than $782 is: A. 0.9079 B. 0.5473 C. 0.0921 D. 0.4527 Thank you!!

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The following analysis involves probability calculations rooted in statistical inference, focusing on proportions and means. The first problem pertains to estimating the probability that the proportion of American coin dealers who believe that investing in Silver Eagles will be profitable next year falls within a specific interval. The second problem involves calculating the probability that the average monthly mortgage payments of home buyers exceed a certain amount based on sample data.

Analyzing the Probability of Coin Dealers’ Beliefs About Silver Eagles

The first problem can be approached using the normal approximation to the binomial distribution. The sample proportion, p̂, follows a distribution that approximates normality when the sample size is sufficiently large, which it is in this case (n=150). Given the population proportion p=0.58, the mean and standard deviation of the sampling distribution of p̂ are calculated as:

  • Mean, μₚ̂ = p = 0.58
  • Standard deviation, σₚ̂ = √[p(1-p)/n] = √[0.58×0.42/150] ≈ √[0.2436/150] ≈ √0.001624 ≈ 0.0403

To find the probability that the proportion falls between 54% and 60%, we convert these percentages to their z-scores:

  • Z for 54%: (0.54 - 0.58)/0.0403 ≈ -0.04/0.0403 ≈ -0.99
  • Z for 60%: (0.60 - 0.58)/0.0403 ≈ 0.02/0.0403 ≈ 0.50

Using standard normal distribution tables or software, the probabilities corresponding to these z-scores are:

  • P(Z
  • P(Z

Therefore, the probability that the proportion is between 54% and 60% is:

0.6915 - 0.1611 = 0.5304

This value aligns most closely with option D, 0.5297.

Analyzing the Probability of Mortgage Payments Exceeding $782

The second problem involves sampling distribution of the mean mortgage payment. The population mean μ= $732 and standard deviation σ= $421 are used with a sample size n=125. Since the sample size is large, the Central Limit Theorem justifies using the normal distribution to approximate the sampling distribution of the sample mean. The standard error (SE) is:

  • SE = σ/√n = 421/√125 ≈ 421/11.1803 ≈ 37.70

To find the probability that the sample mean exceeds $782, we compute the z-score for $782:

  • Z = (X̄ - μ)/SE = (782 - 732)/37.70 ≈ 50/37.70 ≈ 1.33

Using standard normal distribution tables or software, the probability that Z exceeds 1.33 is:

P(Z > 1.33) ≈ 1 - P(Z

This value is closest to option C, 0.0921.

Conclusion

In summary, the probability that between 54% and 60% of coin dealers believe it will be a profitable year is approximately 0.5304, aligning with option D. The probability that the average monthly mortgage payment exceeds $782 is approximately 0.0918, matching option C. These calculations demonstrate effective application of the normal distribution in probability estimations based on sample data, essential for decision-making in financial and investment contexts.

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