Solved on Mar 25, 2024

Find the margin of error for a poll of 130 people, where 64%64\% said they liked dogs, at a 90%90\% confidence level.

STEP 1

1. The percentage of people who said they liked dogs is a sample proportion from the poll.
2. The margin of error can be calculated using the formula for the margin of error of a proportion, which is ME=zp(1p)nME = z \cdot \sqrt{\frac{p(1-p)}{n}}, where zz is the z-score corresponding to the confidence level, pp is the sample proportion, and nn is the sample size.
3. The z-score for a 90%90\% confidence level is approximately 1.6451.645, which corresponds to the critical value for a two-tailed test where the tails (each) contain 5%5\% of the distribution (since 100%90%=10%100\% - 90\% = 10\% and 10%/2=5%10\%/2 = 5\%).

STEP 2

1. Calculate the sample proportion (pp).
2. Find the z-score corresponding to the 90%90\% confidence level.
3. Calculate the margin of error using the formula.

STEP 3

Calculate the sample proportion (pp) using the information given in the poll.
p=number of people who like dogstotal number of people polled=64% of 130130=0.64 p = \frac{\text{number of people who like dogs}}{\text{total number of people polled}} = \frac{64\% \text{ of } 130}{130} = 0.64

STEP 4

Find the z-score corresponding to the 90%90\% confidence level.
The z-score for a 90%90\% confidence level is approximately 1.6451.645.

STEP 5

Calculate the margin of error using the formula ME=zp(1p)nME = z \cdot \sqrt{\frac{p(1-p)}{n}}.
ME=1.6450.64(10.64)130 ME = 1.645 \cdot \sqrt{\frac{0.64(1-0.64)}{130}}

STEP 6

Simplify the calculation inside the square root.
ME=1.6450.640.36130 ME = 1.645 \cdot \sqrt{\frac{0.64 \cdot 0.36}{130}}

STEP 7

Calculate the value inside the square root.
0.640.361300.23041300.001773 \frac{0.64 \cdot 0.36}{130} \approx \frac{0.2304}{130} \approx 0.001773

STEP 8

Take the square root of the calculated value.
0.0017730.0421 \sqrt{0.001773} \approx 0.0421

STEP 9

Multiply the square root by the z-score to find the margin of error.
ME=1.6450.04210.0692 ME = 1.645 \cdot 0.0421 \approx 0.0692

STEP 10

Round the margin of error to three decimal places as requested.
ME0.069 ME \approx 0.069
The margin of error of this poll, at the 90%90\% confidence level, is approximately 0.0690.069.

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