Answer:
a) Point estimate p=0.39
The 95% confidence interval for the population proportion is (0.369, 0.411).
b) Point estimate M=10.9 cigarretes a day.
The 95% confidence interval for the mean is (10.84, 10.96).
c) 27.95 cigarretes per day.
Step-by-step explanation:
<em>The question is incomplete:</em>
<em>The sample data is not attached.</em>
<em>We can work with a random and representative sample where, from 2000 Russians interviewed, 780 are smokers.</em>
<em>Out of this 780 smokers, 550 smoke a package a day, 150 smoke two packages a day and 80 smoke three packages a day. The packages have 20 cigarettes each.</em>
a) We have to calculate a 95% confidence interval for the proportion.
The score is X=780, with a sample size n=2000.
The point estimate for the sample population is the sample proportion and has a value of p=0.39.
![p=X/n=780/2000=0.39](https://tex.z-dn.net/?f=p%3DX%2Fn%3D780%2F2000%3D0.39)
The standard error of the proportion is:
![\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.39*0.61}{2000}}\\\\\\ \sigma_p=\sqrt{0.000119}=0.011](https://tex.z-dn.net/?f=%5Csigma_p%3D%5Csqrt%7B%5Cdfrac%7Bp%281-p%29%7D%7Bn%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B0.39%2A0.61%7D%7B2000%7D%7D%5C%5C%5C%5C%5C%5C%20%5Csigma_p%3D%5Csqrt%7B0.000119%7D%3D0.011)
The critical z-value for a 95% confidence interval is z=1.96.
The margin of error (MOE) can be calculated as:
![MOE=z\cdot \sigma_p=1.96 \cdot 0.011=0.021](https://tex.z-dn.net/?f=MOE%3Dz%5Ccdot%20%5Csigma_p%3D1.96%20%5Ccdot%200.011%3D0.021)
Then, the lower and upper bounds of the confidence interval are:
![LL=p-z \cdot \sigma_p = 0.39-0.021=0.369\\\\UL=p+z \cdot \sigma_p = 0.39+0.021=0.411](https://tex.z-dn.net/?f=LL%3Dp-z%20%5Ccdot%20%5Csigma_p%20%3D%200.39-0.021%3D0.369%5C%5C%5C%5CUL%3Dp%2Bz%20%5Ccdot%20%5Csigma_p%20%3D%200.39%2B0.021%3D0.411)
The 95% confidence interval for the population proportion is (0.369, 0.411).
b) The point estimate for the mean annual per capita consumption of cigarettes can be calculated as:
![M_X=\dfrac{\sum n_iX_i}{n}=\dfrac{1220\cdot0+550\cdot 20+150\cdot 40+80\cdot 60}{2000}\\\\\\M_X=\dfrac{0+11000+6000+4800}{2000}=\dfrac{21800}{2000}=10.9](https://tex.z-dn.net/?f=M_X%3D%5Cdfrac%7B%5Csum%20n_iX_i%7D%7Bn%7D%3D%5Cdfrac%7B1220%5Ccdot0%2B550%5Ccdot%2020%2B150%5Ccdot%2040%2B80%5Ccdot%2060%7D%7B2000%7D%5C%5C%5C%5C%5C%5CM_X%3D%5Cdfrac%7B0%2B11000%2B6000%2B4800%7D%7B2000%7D%3D%5Cdfrac%7B21800%7D%7B2000%7D%3D10.9)
The standard deviation can be calculated as:
![s_X=\sqrt{\dfrac{\sum n_i(X_i-M_x)^2}{n-1}}\\\\\\s_X=\sqrt{\dfrac{1220(0-10.9)^2+550(20-10.9)^2+150(40-10.9)^2+80(60-10.9)^2}{1999}}\\\\\\s_X=\sqrt{\dfrac{118.81+82.81+846.81+2410.81}{1999}}=\sqrt{\dfrac{3459.24}{1999}}=\sqrt{1.73}\approx1.32](https://tex.z-dn.net/?f=s_X%3D%5Csqrt%7B%5Cdfrac%7B%5Csum%20n_i%28X_i-M_x%29%5E2%7D%7Bn-1%7D%7D%5C%5C%5C%5C%5C%5Cs_X%3D%5Csqrt%7B%5Cdfrac%7B1220%280-10.9%29%5E2%2B550%2820-10.9%29%5E2%2B150%2840-10.9%29%5E2%2B80%2860-10.9%29%5E2%7D%7B1999%7D%7D%5C%5C%5C%5C%5C%5Cs_X%3D%5Csqrt%7B%5Cdfrac%7B118.81%2B82.81%2B846.81%2B2410.81%7D%7B1999%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B3459.24%7D%7B1999%7D%7D%3D%5Csqrt%7B1.73%7D%5Capprox1.32)
We have to calculate a 95% confidence interval for the mean.
The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.
The sample mean is M=10.9.
The sample size is N=2000.
When σ is not known, s divided by the square root of N is used as an estimate of σM:
The degrees of freedom for this sample size are:
The t-value for a 95% confidence interval and 1999 degrees of freedom is t=1.961.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
The 95% confidence interval for the mean is (10.84, 10.96).
c. Only for the proportion of smokers, the expected value for the number of cigarretes smoked per day is:
![E(Y)=\dfrac{550\cdot 20+150\cdot 40+80\cdot 60}{780}=\dfrac{21800}{2000}=27.95](https://tex.z-dn.net/?f=E%28Y%29%3D%5Cdfrac%7B550%5Ccdot%2020%2B150%5Ccdot%2040%2B80%5Ccdot%2060%7D%7B780%7D%3D%5Cdfrac%7B21800%7D%7B2000%7D%3D27.95)