Answer: we reject the initial claim
(50. 3802, 51.6198)
Step-by-step explanation:
A)
The initial statement is that the mean temperature throughout history is 50°
This is the null hypothesis and it is denoted as H'
H': u = 50.
After taking a sample of 40 years, we realized that the mean temperature is 51°.
This is the alternative hypothesis, in contradiction to the alternative.
The alternative hypothesis is denoted as H1
H1: u > 50 ( upper tailed).
Sample size (n) = 40
Sample mean (x) = 51
Population standard deviation (σ) = 2
We use a z test to get the value of the test statistics.
We are using a z test because sample size is greater than 30 ( n = 40) and population standard deviation is given.
Z score = x - u/ (σ/√n)
Z score = 51 - 50 / (2/√40)
Z score = 1 / 0.3162
Z score = 3.16
Our level of significance is 5%, and the critical value at this level of significance is 1.645.
By comparing the z score relative to the critical value, since our z score is greater than 1.645, it implies that we are in the rejection region hence we reject the initial claim.
B)
We are to construct a 95% confidence interval for population mean temperature.
For upper limit
u = x + Zα/2 ×(σ/√n)
Where Zα/2 = 1.96 which is the critical value for a two tailed test at 5% level of significance.
For upper limit, we have that
u = 51 + 1.96 × (2/√40)
u = 51 + 1.96 (0.3162)
u = 51 + 0.6198
u = 51.6198.
For lower limit, we have that
u = 51 - 1.96 × (2/√40)
u = 51 - 1.96 (0.3162)
u = 51 - 0.6198
u = 50. 3802.
Hence the 95% confidence level for mean temperature is given as (50. 3802°, 51.6198°)
such that
, and because the distribution is symmetric about its mean, it follows that 
.
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