Exhibit 10-10
The results of a recent poll on the preference of shoppers regarding two products are shown below.

Product

Shoppers Surveyed

Shoppers Favoring
This Product

A

800

560

B

900

612

Refer to Exhibit 10-10. At 95% confidence for the interval estimation of P 1-P 2, the margin of error is _____.

Question

Exhibit 10-10
The results of a recent poll on the preference of shoppers regarding two products are shown below.

Product

Shoppers Surveyed

Shoppers Favoring
This Product

A

800

560

B

900

612

​

Refer to Exhibit 10-10. At 95% confidence for the interval estimation of P 1-P 2, the margin of error is  _____.

BlackTom AI · Accepted Answer

We start by identifying the data given for the two products from Exhibit 10-10. For product A, n1 = 800 shoppers, with 560 favoring it, so p1_hat = 560/800 = 0.70. For product B, n2 = 900 shoppers, with 612 favoring it, so p2_hat = 612/900 = 0.68. The observed difference in sample proportions is p1_hat - p2_hat = 0.70 - 0.68 = 0.02.

Now, to construct a 95% confidence interval for the true difference P1 - P2, we typically use the standard error based on the pooled proportion when performing an interval for a difference of proportions from two independent samples: SE = sqrt( p_hat_pooled * (1 - p_hat_pooled) * (1/n1 + 1/n2) ), where p_hat_pooled = (x1 + x2) / (n1 + n2) = (560 + 612) / (800 + 900) = 1172 / 1700 ≈ 0.6894.

Compute 1 - p_hat_pooled ≈ 0.3106, so p_hat_pooled * (1 - p_hat_pooled) ≈ 0.6894 * 0.3106 ≈ 0.214.

Next, add the reciprocal sample sizes: 1/n1 + 1/n2 = 1/800 + 1/900 = 0.00125 + 0.001111... ≈ 0.002361. Multiply: 0.214 * 0.002361 ≈ 0.000504. Take the square root to get SE ≈ sqrt(0.000504) ≈ 0.0224.

For a 95% confidence interval, use the z-critical value for 95%: z ≈ 1.96. The margin of error is MOE = z * SE ≈ 1.96 * 0.0224 ≈ 0.0439, which rounds to about 0.044.

Now let’s evaluate each answer option:

Option 1: ".044" — This matches the computed MOE of approximately 0.0439, so it aligns with the standard error calculation and z-value used for a 95% interval.

Option 2: "None of the other three" — Claiming none of the other options are correct would require that all other options are wrong. Since we found a calculation that supports smaller MOE values, this would be inconsistent with the numeric result unless all other listed numeric choices were also shown to be wrong, which they aren’t yet evaluated.

Option 3: ".064" — This MOE would occur if SE were around 0.0327 (since 1.96 * 0.0327 ≈ 0.064). That SE is larger than the computed SE, which would require a different SE calculation (for example, using an unpooled standard error or a miscomputed p_hat_pooled), and does not reflect the pooled estimate given the sample sizes and success counts.

Option 4: ".0225" — This is roughly half the computed MOE. An SE of about 0.0115 would be needed to yield 0.0225 as MOE with a 1.96 multiplier, which is not consistent with the pooled-proportion SE given the data. This undervalues the uncertainty by mixing up the standard error or the z-critical value.

In summary, option by option, we see that the value around 0.044 is consistent with the standard pooling approach for two proportions at 95% confidence, while the other numbers do not align with the standard calculation using the provided sample sizes and counts.

类似问题

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.

Questions 8 and 9 continued: Report the upper limit of the confidence interval to three decimal places. Make sure your answer is a difference of proportions, not a difference of percentages.

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.

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类似问题

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 ​ ​ Refer to Exhibit 10-10. The standard error of is _____.

Questions 8 and 9 continued: Report the upper limit of the confidence interval to three decimal places. Make sure your answer is a difference of proportions, not a difference of percentages.

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 ​ ​ Refer to Exhibit 10-10. The standard error of is _____.

更多留学生实用工具

希望你的学习变得更简单

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.