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A telecommunications company wants to estimate the proportion of households that would purchase an additional telephone line if it were made available at a substantially reduced installation cost. Data are collected from a random sample of 500 households. The results indicate that 155 of the households would purchase the additional telephone line at a reduced installation cost. Complete parts (a) and (b) below.

a. Construct a 95% confidence interval estimate for the population proportion of households that would purchase the additional telephone line.

b. The margin of error is = ____ ?

c. The critical value for the upper tail is __ ?

d. How would the manager in charge of promotional programs concerning residential customers use the results in (a)?

i. The manager can infer with 95% confidence that the population proportion of all households that would purchase an additional telephone line is somewhere in
this interval 
ii. The manager can infer with 95% confidence that the proportion of households in the sample that would purchase an additional telephone line is somewhere in
this interval. 
iii. The manager can infer that 95% of households would purchase an additional telephone line. 
iv. The manager can infer that the true population proportion of households that would purchase an additional telephone line is somewhere in this interval 95% of the time.
 

It should be well understandable to everyone this is a question about the test of proportions. Construction of the confidence interval given the sample statistics above is guided by one assumption that is required before conducting inferential analysis using the normal distribution. Confidence interval for proportions are constructed using the standard normal distribution and thus, it is assumed that the responses are normally distributed and the sample selected is a simple random sample from a normally distributed population. The other requirement is that the sample size should be greater than 30 and the number of success and failures in the sample data should be greater than 5. Some writers set the minimum number of successes to be greater than 10. 

it is given from the sample statistics that sampe size is 500 and the number of success are 155,  Therefore, the requirement needed for the number of successes and failures is fullfiled. 

The confidence interval for proportion is constructed using the formula;- p^ +/- Z-score * SE

Standard error for the confidence interval for proportions;- sqrt(p^ * (1-p^)/n)

p_hatch = 155/500 = 0.31

SE = sqrt(0.31 * (1-0.31)/500) = 0.0207

z-score = 1.96

Hence the confidence interval;- 0.31 +/- 1.96 * 0.0207 

95% confidence interval for this proportion is (0.2694, 0.3506)

The margin of error  = 0.0406

c. 

Critical value for the upper tail. 

 The critical value to be calculated depends with what the alternative hypothesis is set to be. If we are testing hypothesis about the test of proportions, then the critical value with be  =NORM.S.INV(0.95) = 1.645, Here, the critical value for the z-score is calculated in excel. 

and so if one is conducting test of hypothesis using this, they would be required to validate the alternative hypothesis if the z-score is greater than 1.645. 

d. 

interpretations  for the confidence intervals for proportions is similar to the other inferential tests like t-statistic. Hence, the interpretation follows that one is 95% confident that the true population proportion of the event of interest lies within the lower and the upper bounds of the confidence interval found. 

 

 

 

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