Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
1) n = 128, x = 93; 90 percent
To construct a confidence interval for the population proportion, we can use the formula:
Confidence Interval = sample proportion ± margin of error
where the sample proportion is denoted by "p̂" and the margin of error is determined using the formula:
Margin of Error = critical value * standard error
First, let's calculate the sample proportion (p̂):
Sample Proportion (p̂) = x / n
Given that x = 93 and n = 128, we can calculate:
p̂ = 93 / 128 = 0.7266
Next, we need to find the critical value associated with the given degree of confidence. For a 90% confidence level, we need to find the z-value that leaves 5% in each tail. Using a z-table or a statistical calculator, we find that the z-value associated with a 90% confidence level is approximately 1.645.
Now, we can calculate the standard error using the formula:
Standard Error = sqrt( (p̂ * (1 - p̂)) / n )
Standard Error = sqrt( (0.7266 * (1 - 0.7266)) / 128 )
Standard Error ≈ 0.039
Next, we can calculate the margin of error using the formula:
Margin of Error = critical value * standard error
Margin of Error = 1.645 * 0.039 ≈ 0.064
Finally, we can construct the confidence interval by adding and subtracting the margin of error from the sample proportion:
Confidence Interval = p̂ ± margin of error
Confidence Interval = 0.7266 ± 0.064
Confidence Interval ≈ (0.6636, 0.7896)
Therefore, using the given degree of confidence (90%) and sample data (n = 128, x = 93), the confidence interval for the population proportion p is approximately (0.6636, 0.7896).
To construct a confidence interval for the population proportion, we can use the following formula:
CI = p̂ ± Z * √(p̂(1-p̂) / n)
where:
p̂ is the sample proportion (x/n),
Z is the critical value from the standard normal distribution corresponding to the desired level of confidence, and
n is the sample size.
Given:
n = 128 (sample size)
x = 93 (number of successes)
Confidence level = 90 percent
First, we need to find the critical value (Z) for a 90 percent confidence interval. The critical value is determined based on the desired level of confidence. For a 90 percent confidence level, we need to find the Z-value such that the area under the normal distribution curve between -Z and Z is 0.90.
Using a standard normal distribution table or a statistical software, we find that the critical Z-value for a 90 percent confidence level is approximately 1.645.
Next, we can substitute the given values into the formula:
CI = p̂ ± Z * √(p̂(1-p̂) / n)
CI = 93/128 ± 1.645 * √((93/128)(1-(93/128)) / 128)
Calculating the values inside the square root:
CI = 93/128 ± 1.645 * √((93/128)(35/128) / 128)
CI = 93/128 ± 1.645 * √(324.075 / 128)
CI = 93/128 ± 1.645 * √(2.53103)
CI = 93/128 ± 1.645 * 1.5896
CI = 0.7266 ± 2.6155
The confidence interval for the population proportion (p) is approximately 0.7266 ± 0.2616 (rounded to four decimal places).
Therefore, the confidence interval is (0.465, 0.988) at a 90 percent confidence level.