Find alpha from CI width

3 A student wishes to estimate the proportion, \(p\), of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate \(\alpha \%\) confidence interval for \(p\) and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures. Find \(\alpha\) correct to the nearest integer.

3 A researcher wishes to estimate the proportion, \(p\), of houses in London Road that have only one occupant. He takes a random sample of 64 houses in London Road and finds that 8 houses in the sample have only one occupant. Using this sample, he calculates that an approximate \(\alpha \%\) confidence interval for \(p\) has width 0.130 . Find \(\alpha\) correct to the nearest integer.

3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.

2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha \%\) confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer.

1. The random variable \(X\) is normally distributed with unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\)

A random sample of 25 observations of \(X\) produced a \(95 \%\) confidence interval for \(\mu\) of (26.624, 28.976)

1. Find the mean of the sample.
2. Show that the standard deviation is 3 The \(a\) \% confidence interval using the 25 observations has a width of 2.1
3. Calculate the value of \(a\)
4. Find the smallest sample size, of observations from \(X\), that would be required to obtain a 95\% confidence interval of width at most 1.5

Each of a random sample of 80 adults gave an estimate, \(h\) metres, of the height of a particular building. The results were summarised as follows. $$n = 80 \quad \sum h = 2048 \quad \sum h^2 = 52760$$

1. Calculate unbiased estimates of the population mean and variance. [3]
2. Using this sample, the upper boundary of an \(\alpha\%\) confidence interval for the population mean is 26.0. Find the value of \(\alpha\). [4]

In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha\)% confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487, correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer. [4]