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 to 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

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u= 300
Alternative hypothesis: u 300

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the sample mean is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 2.041

b)

DF = n - 1

D.F = 3

t_critical = + 3.182

c)

t = (x - u) / SE

t = - 1.225

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

Since we have a two-tailed test, the P-value is the probability that the t statistic having 3 degrees of freedom is less than -1.225 or greater than 1.225.

Thus, the P-value = 0.308.

d)

Interpret results. Since the P-value (0.308) is greater than the significance level (0.05), we cannot reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that the dial is properly calibrated.