A genetic experiment with peas resulted in one sample of offspring that of 421 green peas 155 yellow peas. Construct a 95% conf. Interval for yellow peas. _<p<_
To construct a 95% confidence interval for the proportion of yellow peas, we can use the formula:
Confidence Interval = sample proportion ± (critical value * standard error)
First, let's calculate the sample proportion of yellow peas:
Sample Proportion (p) = Number of yellow peas / Total number of peas
= 155 / (421 + 155)
≈ 0.269
Next, we need to determine the critical value associated with a 95% confidence level. For a large enough sample size (which we assume to be in this case), the critical value can be calculated using the Z-distribution or standard normal distribution table. For a 95% confidence level, the critical value is approximately 1.96.
Standard Error (SE) can be calculated using the formula:
SE = √((p * (1 - p)) / n)
Where,
p = Sample proportion
n = Total number of peas
Now, let's use these values to calculate the confidence interval:
Confidence Interval = 0.269 ± (1.96 * √((0.269 * (1 - 0.269)) / (421 + 155)))
Calculating the standard error:
SE = √((0.269 * (1 - 0.269)) / (421 + 155))
≈ 0.024
Finally, we can substitute the values into the confidence interval formula:
Confidence Interval = 0.269 ± (1.96 * 0.024)
= 0.269 ± 0.047
Therefore, the 95% confidence interval for the proportion of yellow peas is approximately 0.222 to 0.316.
To construct a 95% confidence interval for the number of yellow peas, we can use the formula:
Confidence Interval = Sample Proportion ± (Critical Value) * √((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
First, we need to determine the sample proportion of yellow peas:
Sample Proportion = (Number of Yellow Peas) / (Total Sample Size)
Sample Proportion = 155 / (421 + 155)
Sample Proportion ≈ 0.269
Next, we need to find the critical value for a 95% confidence interval. Since we don't have the exact sample size mentioned in the question, we can assume it to be large enough for the normal approximation. In this case, for a 95% confidence interval, the critical value can be obtained from a standard normal distribution table or calculator.
The critical value for a 95% confidence interval is approximately 1.96.
Now, we can plug in the values into the formula:
Confidence Interval = 0.269 ± 1.96 * √((0.269 * (1 - 0.269)) / (421 + 155))
Calculating the confidence interval using the given formula, we find:
Confidence Interval ≈ 0.269 ± 1.96 * √((0.269 * 0.731) / (576))
Simplifying the equation:
Confidence Interval ≈ 0.269 ± 1.96 * √((0.196501) / (576))
Calculating the square root and simplifying further:
Confidence Interval ≈ 0.269 ± 1.96 * 0.01400797
Finally, calculating the upper and lower bounds of the confidence interval:
Lower Bound ≈ 0.269 - (1.96 * 0.01400797)
Upper Bound ≈ 0.269 + (1.96 * 0.01400797)
Therefore, the 95% confidence interval for the number of yellow peas is approximately:
0.24 < p < 0.30, where "p" represents the proportion of yellow peas.