A bag containing 10,000 beads of assorted colors is purchased from a craft store. To estimate the percent of red beads in the bag, a sample of beads is selected at random. The percent of red beads in the bag was estimated to be 15%, with an associated margin of error of 2%. If r is the actual number of red beads in the bag, which of the following is most plausible?

The percent of red beads in the bag is estimated to be 15% with a margin of error of 2%, we need to determine the most plausible range for the actual number of red beads, r, in the bag.

let’s calculate the range of the percentage of red beads in the bag considering the margin of error.
→ The estimated percent is 15%.
→ The margin of error is ±2%.

Thus, the actual percentage of red beads could range from :
15% − 2% = 13% to 15% + 2% = 17%.

Now, let’s convert these percentages to the actual number of red beads in a bag of 10,000 beads:
→ For the lower bound (13%):
\$13/100 times 10000\$ = 1300
→ For the upper bound (17%):
\$17/100 times 10000\$ = 1700

Therefore, the number of red beads r is most likely to be in the range is 1,300 ≤ r ≤ 1,700.

This option suggests that the number of red beads is greater than 1,700. Since our estimated upper bound is 1,700, it is unlikely that the actual number of red beads exceeds this value given the margin of error. Thus, this option is not plausible

r > 1,700

This option indicates that the number of red beads is strictly between 1,300 and 1,700. This range falls within the estimated bounds (1,300 to 1,700), suggesting it is a plausible estimate of the actual number of red beads.Therefore, this option is plausible

1,300 < r < 1,700

This option suggests that the number of red beads is between 200 and 1,500. While part of this range (1,300 to 1,500) overlaps with our estimated range, the lower bound of 200 is far below our calculated lower bound of 1,300. This implies that the actual percentage of red beads could be as low as 2%, which is not consistent with our estimated percentage range of 13% to 17%

200 < r < 1,500

This option suggests that the number of red beads is less than 1,500. While it includes part of our estimated range (1,300 to 1,500), it excludes the upper part of our range (1,500 to 1,700). This implies that the actual percentage of red beads could be below 15%, but not extending up to 17%, which is inconsistent with the full range we calculated

r < 1,500

Option

B

Can you summarize what you’ve understood in the above steps?