In sample of 1000 TV viewers, 330 watched a particular programme. Find 99% confidence limits for TV viewers who watch this programme .
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Well, it seems like the TV viewers who watched this particular program were quite the selective bunch. Maybe it was just that gripping storyline or the enticing characters. Now, to find the confidence limits, we need to take a bit of a statistical detour.
To calculate the confidence limits, we'll be using the formula:
Confidence Limits = Sample Proportion ± Margin of Error
The sample proportion, in this case, is the proportion of viewers who watched the program. So, we divide the number of viewers who watched (330) by the total sample size (1000) to get 0.33.
Now let's calculate the margin of error, shall we? Since we want a 99% confidence level, we'll be using a z-score of 2.576 (which corresponds to the 99% confidence level). The formula for the margin of error is:
Margin of Error = z * sqrt((p * (1-p)) / n)
Plugging in the values:
Margin of Error = 2.576 * sqrt((0.33 * (1-0.33)) / 1000)
After some calculations, we get a margin of error of approximately 0.022.
Now, we can find the confidence limits by adding and subtracting the margin of error to the sample proportion:
Lower Limit = 0.33 - 0.022 = 0.308
Upper Limit = 0.33 + 0.022 = 0.352
So, with 99% confidence, we can say that the proportion of TV viewers who watch this program lies between approximately 30.8% and 35.2%. But hey, don't worry if the numbers aren't exactly what you expected. Statistics is a bit like a clown juggling balls – sometimes it's spot-on, and sometimes it misses the mark!
To find the 99% confidence limits for TV viewers who watch the program, we can use the formula for the confidence interval for proportions:
CI = p̂ ± z * √(p̂ * (1 - p̂) / n)
Where:
CI = Confidence Interval
p̂ = Sample Proportion
z = Z-score
n = Sample Size
Given that 330 out of 1000 viewers watched the program, the sample proportion (p̂) is 330/1000 = 0.33.
To find the Z-score for a 99% confidence level, we need to find the corresponding Z-value from a Z-table or use a calculator. The Z-score for a 99% confidence level is approximately 2.576.
Now we can plug in the values into the formula:
CI = 0.33 ± 2.576 * √(0.33 * (1 - 0.33) / 1000)
Calculating the square root and multiplying it with the Z-score gives us:
√(0.33 * (1 - 0.33) / 1000) ≈ 0.0153
Finally, we can calculate the confidence interval:
CI = 0.33 ± 2.576 * 0.0153
This gives us two confidence limits:
Lower Limit: 0.33 - (2.576 * 0.0153) ≈ 0.328
Upper Limit: 0.33 + (2.576 * 0.0153) ≈ 0.332
Therefore, the 99% confidence interval for the proportion of TV viewers who watch the program is approximately 0.328 to 0.332.
To find the 99% confidence limits for TV viewers who watched the particular program, we can use the formula for calculating confidence intervals for proportions.
The formula for the confidence interval is:
CI = p̂ ± z * √((p̂ * (1 - p̂)) / n)
Where:
- CI is the confidence interval
- p̂ is the sample proportion (330/1000 = 0.33 in this case)
- z is the z-value corresponding to the desired confidence level (in this case, 99% confidence level)
- n is the sample size (1000 in this case)
First, we need to find the z-value for a 99% confidence level. The z-value can be found using a z-table or a statistical calculator. For a 99% confidence level, the z-value is approximately 2.576.
Now, we can substitute the values into the formula:
CI = 0.33 ± 2.576 * √((0.33 * (1 - 0.33)) / 1000)
Calculating the expression within the square root:
√((0.33 * (1 - 0.33)) / 1000) ≈ √(0.2211 / 1000) ≈ √0.0002211 ≈ 0.01488
Substituting back into the formula:
CI = 0.33 ± 2.576 * 0.01488
Calculating the upper and lower limits of the confidence interval:
Upper limit = 0.33 + (2.576 * 0.01488) ≈ 0.33 + 0.0383 ≈ 0.3683
Lower limit = 0.33 - (2.576 * 0.01488) ≈ 0.33 - 0.0383 ≈ 0.2917
Therefore, the 99% confidence limits for TV viewers who watched the particular program are approximately 29.17% to 36.83%.