# With 90 % confidence, for sample mean 319.00, sample standard deviation 13.80, and sample size...

## Question:

With {eq}90 \% {/eq} confidence, for sample mean {eq}319.00 {/eq}, sample standard deviation {eq}13.80 {/eq}, and sample size {eq}35 {/eq}, what is the upper confidence limit with 2 decimal places?

## Upper confidence limit

Upper confidence limit of the mean is computed while finding the confidence interval. It includes:-

i. Sample mean,

ii. Critical value and

iii. Standard error of the mean

Given that sample mean, {eq}\bar{X} {/eq} = 319, sample standard deviation, {eq}s {/eq} = 13.80, sample size, {eq}n {/eq} = 35 and level of confidence, {eq}\alpha {/eq} = 1 - 0.90 = 0.10

Since the population standard deviation is unknown, we use t-test.

First, we find the critical value at degrees of freedom, {eq}df {/eq} = n - 1 = 35 - 1 = 34

Using t-tables, the critical value is

{eq}t (\alpha, df) = t (0.1,34) {/eq} = 1.307

The upper limit of the confidence interval is given by:-

{eq}\bar{X} + t\times s/\sqrt{n} {/eq}

{eq}319 + 1.307\times 13.80/\sqrt{35} {/eq}

319 + 1.307 \times 2.333

319 + 3.05

322.05 