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

Studdy Solution

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=z⋅p(1−p)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=z⋅p(1−p)nME = z \cdot \sqrt{\frac{p(1-p)}{n}}.
ME=1.645⋅0.64(1−0.64)130 ME = 1.645 \cdot \sqrt{\frac{0.64(1-0.64)}{130}}

STEP 6

Simplify the calculation inside the square root.
ME=1.645⋅0.64⋅0.36130 ME = 1.645 \cdot \sqrt{\frac{0.64 \cdot 0.36}{130}}

STEP 7

Calculate the value inside the square root.
0.64⋅0.36130≈0.2304130≈0.001773 \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.
0.001773≈0.0421 \sqrt{0.001773} \approx 0.0421

STEP 9

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

STEP 10

Round the margin of error to three decimal places as requested.
ME≈0.069 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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