Answer:
the minimum sample size required is 139
Explanation:
To construct a 95% confidence interval with a maximum error of 0.19 reproductions per hour, we need to use the t-distribution. The t-distribution is used when the population variance is unknown and the sample size is small.
To find the minimum sample size required, we need to use the following formula:
n = (t_(α/2,df) * s / E)^2
where:
n is the minimum sample size
t_(α/2,df) is the critical value of the t-distribution for a confidence level of 95% (α = 0.05) and df is the degrees of freedom
s is the standard deviation of the population
E is the maximum error allowed
We are given that the mean is 12.6 reproductions and the variance is 3.61. The standard deviation is the square root of the variance, so the standard deviation is 1.9.
To find the critical value of the t-distribution, we need to know the degrees of freedom. The degrees of freedom is equal to the sample size minus 1. Since we don't know the sample size yet, we will use a symbol (let's use "df") to represent the degrees of freedom in the formula.
Plugging the values into the formula, we get:
n = (t_(0.025,df) * 1.9 / 0.19)^2
To find the critical value of the t-distribution, we need to use a t-table or a computer program. Looking up the critical value in a t-table or using a computer program, we find that the critical value for a 95% confidence level and df = 9 is 2.262.
Plugging this value into the formula, we get:
n = (2.262 * 1.9 / 0.19)^2
Simplifying the expression, we get:
n = (11.868)^2
n = 138.7
The minimum sample size required is 138.7. Since we can't have a fractional number of samples, we need to round up to the next integer, which is 139.
Therefore, the minimum sample size required is 139.