Answer:
The 90% confidence interval for the difference of the population means is approximately (-17.98, -2.02).
Step-by-step explanation:
Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
and standard deviation ![s = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Boys:
Mean of 75, sample of 45, standard deviation of 25.
This means that ![\mu_B = 75, s_B = \frac{25}{\sqrt{45}}](https://tex.z-dn.net/?f=%5Cmu_B%20%3D%2075%2C%20s_B%20%3D%20%5Cfrac%7B25%7D%7B%5Csqrt%7B45%7D%7D)
Girls:
Mean of 85, sample of 30, standard deviation of 17.
This means that ![\mu_G = 85, s_G = \frac{17}{\sqrt{30}}](https://tex.z-dn.net/?f=%5Cmu_G%20%3D%2085%2C%20s_G%20%3D%20%5Cfrac%7B17%7D%7B%5Csqrt%7B30%7D%7D)
Distribution of the difference of mean grades of boys and girls:
![\mu = \mu_B - \mu_G = 75 - 85 = -10](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cmu_B%20-%20%5Cmu_G%20%3D%2075%20-%2085%20%3D%20-10)
![s = \sqrt{s_B^2+s_G^2} = \sqrt{(\frac{25}{\sqrt{45}})^2+(\frac{17}{\sqrt{30}})^2} = 4.85](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_B%5E2%2Bs_G%5E2%7D%20%3D%20%5Csqrt%7B%28%5Cfrac%7B25%7D%7B%5Csqrt%7B45%7D%7D%29%5E2%2B%28%5Cfrac%7B17%7D%7B%5Csqrt%7B30%7D%7D%29%5E2%7D%20%3D%204.85)
Confidence interval:
As stated, the critical value is ![z = 1.645](https://tex.z-dn.net/?f=z%20%3D%201.645)
The margin of error is of:
![M = zs = 1.645*4.85 = 7.98](https://tex.z-dn.net/?f=M%20%3D%20zs%20%3D%201.645%2A4.85%20%3D%207.98)
Lower bound:
![\mu - M = -10 - 7.98 = -17.98](https://tex.z-dn.net/?f=%5Cmu%20-%20M%20%3D%20-10%20-%207.98%20%3D%20-17.98)
Upper bound:
![\mu + M = -10 + 7.98 = -2.02](https://tex.z-dn.net/?f=%5Cmu%20%2B%20M%20%3D%20-10%20%2B%207.98%20%3D%20-2.02)
The 90% confidence interval for the difference of the population means is approximately (-17.98, -2.02).