Answer to Question #232077 in Statistics and Probability for tam

Question #232077

(a) A machine is used to fill soft drink in bottles. 35 randomly selected bottles

are taken from the output of the machine. Those bottles have a mean

volume of 330 ml with a standard deviation of 4.8 ml. Construct a 98%

confidence interval to estimate the true mean volume of soft drink.

(5 marks)

(b) The marketing department of a smart phone store is interested in

determining the proportion of new smart phone owners who would have

spent more than or equal to RM 2000 to buy a new smart phone. Suppose

that a survey of 45 new customers, 23 indicate that they would have

purchased new smart phone which cost more than or equal to RM2000.

Construct a 99% confidence interval for the true proportion of new smart

phone owners who have spent more than or equal to RM2000 for a new

smartphone. (5 marks)


1
Expert's answer
2022-02-21T17:35:24-0500

(a) The critical value for α=0.02\alpha = 0.02 and df=n1=34df = n-1 = 34 degrees of freedom is tc=z1α/2;n1=2.44115.t_c = z_{1-\alpha/2; n-1} = 2.44115.

The corresponding confidence interval is computed as shown below:


CI=(Xˉtc×sn,Xˉ+tc×sn)CI=(\bar{X}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{X}+t_c\times\dfrac{s}{\sqrt{n}})

=(3302.44115×4.835,=(330-2.44115\times\dfrac{4.8}{\sqrt{35}},

330+2.44115×4.835)330+2.44115\times\dfrac{4.8}{\sqrt{35}})

Therefore, based on the data provided, the 98% confidence interval for the population mean is 328.0194<μ<331.9806,328.0194<\mu<331.9806, which indicates that we are 98% confident that the true population mean μ\mu is contained by the interval (328.0194,331.9806).(328.0194, 331.9806).


(b) The sample proportion is computed as follows, based on the sample size N=45N = 45 and the number of favorable cases X=23:X=23:


p^=X/N=23/45\hat{p}=X/N=23/45

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}})

=(23452.57582345(12345)45,=\bigg(\dfrac{23}{45}-2.5758\sqrt{\dfrac{\dfrac{23}{45}(1-\dfrac{23}{45})}{45}},

2345+2.57582345(12345)45)\dfrac{23}{45}+2.5758\sqrt{\dfrac{\dfrac{23}{45}(1-\dfrac{23}{45})}{45}}\bigg)

=(0.3192,0.7031)=(0.3192, 0.7031)

Therefore, based on the data provided, the 99% confidence interval for the population proportion is 0.3192<p<0.7031,0.3192 < p < 0.7031, which indicates that we are 99% confident that the true population proportion pp is contained by the interval (0.3192,0.7031).(0.3192, 0.7031).



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