Question #336351

In a survey of 300 random households in a particular village, mother's have been asked if they used cellphone to communicate there are 76 who said YES, Use 90%, 95% and 99% confidence to estimate the proportion of all mothers who use the cellphone to communicate

1
Expert's answer
2022-05-02T16:33:28-0400

The sample proportion is computed as follows, based on the sample size N=300N = 300 and the number of favorable cases X=76:X = 76:

p^=XN=763000.253333\hat{p}=\dfrac{X}{N}=\dfrac{76}{300}\approx0.253333

1. The critical value for α=0.10\alpha = 0.10 is zc=z1α/2=1.6449.z_c = z_{1-\alpha/2} = 1.6449.

The corresponding confidence interval is computed as shown below:


CI(Proportion)=(p^zcp^(1p^)n,CI(Proportion)=(\hat{p}-z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},

p^+zcp^(1p^)n)\hat{p}+z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})

=(763001.644976300(176300)300,=(\dfrac{76}{300}-1.6449\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}},

76300+1.644976300(176300)300)\dfrac{76}{300}+1.6449\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}})

=(0.212,0.295)=(0.212,0.295)

Therefore, based on the data provided, the 90% confidence interval for the population proportion is 0.212<p<0.295,0.212 < p < 0.295, which indicates that we are 90% confident that the true population proportion pp is contained by the interval (0.212,0.295).(0.212, 0.295).


2. The critical value for α=0.05\alpha = 0.05 is zc=z1α/2=1.96.z_c = z_{1-\alpha/2} = 1.96.

The corresponding confidence interval is computed as shown below:


CI(Proportion)=(p^zcp^(1p^)n,CI(Proportion)=(\hat{p}-z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},

p^+zcp^(1p^)n)\hat{p}+z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})

=(763001.9676300(176300)300,=(\dfrac{76}{300}-1.96\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}},

76300+1.9676300(176300)300)\dfrac{76}{300}+1.96\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}})

=(0.204,0.303)=(0.204,0.303)

Therefore, based on the data provided, the 95% confidence interval for the population proportion is 0.204<p<0.303,0.204 < p < 0.303, which indicates that we are 95% confident that the true population proportion pp is contained by the interval (0.204,0.303).(0.204, 0.303).


3. The critical value for α=0.01\alpha = 0.01 is zc=z1α/2=2.5758.z_c = z_{1-\alpha/2} = 2.5758.

The corresponding confidence interval is computed as shown below:


CI(Proportion)=(p^zcp^(1p^)n,CI(Proportion)=(\hat{p}-z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},

p^+zcp^(1p^)n)\hat{p}+z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})

=(763002.575876300(176300)300,=(\dfrac{76}{300}-2.5758\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}},

76300+2.575876300(176300)300)\dfrac{76}{300}+2.5758\sqrt{\dfrac{\dfrac{76}{300}(1-\dfrac{76}{300})}{300}})

=(0.189,0.318)=(0.189,0.318)

Therefore, based on the data provided, the 99% confidence interval for the population proportion is 0.189<p<0.318,0.189< p < 0.318, which indicates that we are 99% confident that the true population proportion pp is contained by the interval (0.189,0.318).(0.189, 0.318).



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