Question

We wish to see if the dial indicating the oven temperature for a certain model oven is properly calibrated. Four ovens of this model are selected at random. The dial on each is set to 300°F, and, after one hour, the actual temperature of each is measured. The temperatures measured are 305°, 310°, 300°, and 305°. Assuming that the actual temperatures for this model when the dial is set for 300° are Normally distributed with mean μ, carry out a hypothesis test t answer: whether the dial is properly calibrated.
a) State the appropriate null and alternative hypothesis.
b) State the critical value for this test.
c) Compute the test statistic and p-value. Round your answer into 3 decimal.
d) Statistical decision by either p-value or rejection region, and conclusion in English.
e) To compute the 98.5% confidence interval, determine what proportion value should be entered into the
function InvT()?
f) To make the length of a 95% confidence interval shorter than 3°, we need at least what sample size?

Answer 1

(a)

Null Hypothesis : H₀: The dialis properly caliberated. = 300

ALternative Hypothesis : H_a : the dial is not properly caliberated . 300

(b) Here for dF = 4 -1 = 3

t_critical = 3.18

(c) Here sample mean 305

standard deviation s = 4.08

standard error of sample mean = 4.08/sqrt(4) = 2.04

Tes statistic

t = (300 - 305)/2.04 = -2.451

p- value = 2 * Pr(t > -2.451, 3) = 0.092

(d) Here as p - value is greater than 0.05, so we failed to reject the null hypothesis and conclude that dial is propoerly caliberated.

(e) 98.5 % confidence interval

Here we will use TINV(0.015,3) = 5.047

we put the proportion value = (1 - 0.985)/2 = 0.0075

(f) Here sasmple size required = n

Width of confidenceinterval = 3

95% confidenceinterval so critical value of t would be atleast 1.96

so here

3/2 = 1.96 * 4.08/sqrt(n)

sqrt(n) = (1.96 * 4.08)/(3/2)

n = 28.42

or n = 29