Answer:
(a) Null Hypothesis, 
: 
41
Alternate Hypothesis, 
: > 41
(b) The value of z test statistics is 1.08.
(c) We conclude that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41.
Step-by-step explanation:
We are given that in a random sample of 40 such periods from Spanish colonial times, the sample mean is x¯ = 47.0. Previous studies of sunspot activity during this period indicate that σ = 35.
It is thought that for thousands of years, the mean number of sunspots per 4-week period was about µ = 41.
Let = <u><em>mean sunspot activity during the Spanish colonial period.</em></u>
(a) Null Hypothesis, : 41 {means that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41}
Alternate Hypothesis, : > 41 {means that the mean sunspot activity during the Spanish colonial period was higher than 41}
The test statistics that would be used here <u>One-sample z test statistics</u> as we know about the population standard deviation;
T.S. = 
~ N(0,1)
where, 
= sample mean = 47
σ = population standard deviation = 35
n = sample of periods from Spanish colonial times = 40
So, <em><u>the test statistics</u></em> = 
= 1.08
(b) The value of z test statistics is 1.08.
(c) <u>Now, the P-value of the test statistics is given by;</u>
P-value = P(Z > 1.08) = 1 - P(Z < 1.08)
= 1 - 0.8599 = <u>0.1401</u>
Since, the P-value of the test statistics is higher than the level of significance as 0.1401 > 0.05, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which <u>we fail to reject our null hypothesis</u>.
Therefore, we conclude that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41.
