A research firm conducted a survey to determine the mean amount (in pesos) heavy smokers spends on cigarettes in a week. A sample of 36 heavy smokers revealed that sample mean is 1000 with the standard deviation of 200. What would be a 95% margin of error for your estimate?
In a December 2016 survey by Pulse Asia, 85% of the 1450 adults disagreed with the view that martial law is needed to resolve the various problems currently facing the country. Use this information to give a point estimate and a 95% margin of error for the true proportion of all Filipinos who believe martial law is not needed to solve national problems.
In a December 2016 survey by Pulse Asia, 85% of the 1450 adults disagreed with the view that martial law is needed to resolve the various problems currently facing the country. Use this information to give a point estimate and a 95% margin of error for the true proportion of all Filipinos who believe martial law is not needed to solve national problems.
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Answer:
For the first question:
The 95% margin of error for estimating the mean amount heavy smokers spend on cigarettes in a week is 67.6 pesos.
For the second question:
The point estimate for the true proportion of all Filipinos who believe martial law is not needed is 85%, with a 95% margin of error of 2.2%.
Step-by-step explanation:
For the first question:
To calculate the margin of error for the mean amount heavy smokers spend on cigarettes, we can use the formula:
Margin of Error = Critical value * (Standard deviation / sqrt(sample size))
Given that the sample mean is 1000 pesos, the standard deviation is 200 pesos, and the sample size is 36, we can calculate the margin of error.
First, we need to find the critical value for a 95% confidence level. The critical value corresponds to the level of confidence and the sample size. Since the sample size is 36, we have 35 degrees of freedom (n-1) for a t-distribution.
Using a t-distribution table or a statistical calculator, the critical value for a 95% confidence level with 35 degrees of freedom is approximately 2.028.
Now we can calculate the margin of error:
Margin of Error = 2.028 * (200 / sqrt(36))
= 2.028 * (200 / 6)
= 67.6 pesos
Therefore, the 95% margin of error for estimating the mean amount heavy smokers spend on cigarettes in a week is 67.6 pesos.
For the second question:
To estimate the true proportion of all Filipinos who believe martial law is not needed to solve national problems, we can use the formula for the margin of error for a proportion:
Margin of Error = Critical value * sqrt((p_hat * q_hat) / n)
where p_hat is the sample proportion (85% or 0.85), q_hat is 1 - p_hat (0.15), and n is the sample size (1450).
To find the critical value for a 95% confidence level, we use the Z-distribution since the sample size is large. The critical value for a 95% confidence level is approximately 1.96.
Now we can calculate the margin of error:
Margin of Error = 1.96 * sqrt((0.85 * 0.15) / 1450)
≈ 0.022 or 2.2%
Therefore, the point estimate for the true proportion of all Filipinos who believe martial law is not needed is 85%, with a 95% margin of error of 2.2%.
❀Dunno❀
To calculate the margin of error for the estimate, we can use the formula:
Margin of Error = Critical value * (Standard deviation / √n)
Given:
Sample size (n) = 36
Sample mean = 1000
Standard deviation = 200
For a 95% confidence level, the critical value can be found using a t-distribution with (n-1) degrees of freedom. Since the sample size is 36, the degrees of freedom is 36 - 1 = 35.
Using a t-table or statistical software, the critical value for a 95% confidence level and 35 degrees of freedom is approximately 2.03.
Substituting the values into the formula, we have:
Margin of Error = 2.03 * (200 / √36)
= 2.03 * (200 / 6)
≈ 67.67
Therefore, the 95% margin of error for the estimate of the mean amount heavy smokers spend on cigarettes in a week is approximately 67.67 pesos.
2. For the proportion of Filipinos who believe martial law is not needed:
To calculate the point estimate and the margin of error for the true proportion, we can use the formula:
Margin of Error = Critical value * √(p_hat * (1 - p_hat) / n)
Given:
Sample size (n) = 1450
Proportion (p_hat) = 0.85
For a 95% confidence level, the critical value can be obtained using a normal distribution. Since the sample size is large (n > 30), we can use the Z-table for the critical value. For a 95% confidence level, the critical value is approximately 1.96.
Substituting the values into the formula, we have:
Margin of Error = 1.96 * √(0.85 * (1 - 0.85) / 1450)
≈ 0.016
Therefore, the point estimate for the true proportion of Filipinos who believe martial law is not needed is 0.85, and the 95% margin of error is approximately 0.016.