Learn Significance Test with Examples & Questions

A matched pairs design was used to compare test scores in a statistics course. Twenty-four students were chosen randomly from a large class. For each chosen student, their test 1 and test 2 scores were recorded. A 95% confidence interval for the difference in test scores was constructed, along with the corresponding 2-sided p-value was given as [−0.70,10.45], and the 2-sided p-value =0.084.
How does one determine whether or not this result is statistically significant at the 5% level of significance?
Also, if a 90 confidence interval was constructed instead of a 95% one, would it contain 0?
Would appreciate some clues.

I hope you are all well.
I am struggling to understand a question I need doing as part of my work.
Specifically, I am asked:
A company has developed a new range of batteries. They test a sample of 50 of these batteries and their mean lifetime to be 7.1 hours. Their older range of batteries followed a normal distribution with mean 7 hours and standard deviation 0.35 hours. Test the claim made by the company that the new range of batteries are longer lasting at the 5 and 1 significance levels.
Is it correct to say the normal distribution for the old range of batteries can be written as $pN(7,{0.35}^2)$ and a Z correspondence equaling $\frac{x-<!--\; -\; -->7}{0.35}$?
Likewise, is it correct to say the new range of batteries follows a normal distribution $pN(7.1,{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$ and a Z correspondence equaling $\frac{50-<!--\; -\; -->7.1}{\mathrm{\sigma <!--\; \sigma \; -->}}$?
I am struggling to know how to incorporate the Significance Levels into the question as well.
Thank you for your help.

An electronics retailer would like to investigate the relationship between the selling price of a digital camera and the demand for it. The table shown below gives the weekly demand for the camera in one particular market along with the corresponding price. These data have a sample correlation​ coefficient, rounded to three decimal​ places, of negative 0.888. Using alpha equals 0.10​, test if the population correlation coefficient between the selling price and the demand for the camera is less than zero.
$\overline{)\begin{array}{cc}\text{Demand}& \text{Price}\\ 17& 300\\ 20& 310\\ 15& 320\\ 11& 330\\ 9& 340\end{array}}$
What is the​ p-value?

A new cream that advertises 52% of women over the age of 50 will experience skin improvement from using their product. A study was conducted in which a sample of 170 women over the age of 50 used the new cream for 6 months. Of those 170 women, 76 of them reported skin improvement (as judged by a dermatologist). Is this evidence that the cream will improve the skin of less than 52 % of women over the age of 50? Perform a hypothesis test using the 0.05 level of significance.
Check the conditions needed to use a sampling distribution (Normal)

The local school board has instituted a book rewards program intended to encourage third graders to read more, thereby increasing their scores on a standardized vocabulary test. Nationally, the average score on this exam among third graders is 98. A random sample of 223 local third graders participating in the program scored an average of 104 on the test with a standard deviation of 13.
Conduct a significance test with a=.05 to answer the research question: Is the program working to significantly increase local third graders' vocabulary scores?
STEP ONE
In step one of this significance test, do you have to have a "random sample" or "independent random samples"?
a) a random sample
b) independent random samples

A manufacturer of sports equipment has developed a new synthetic fighting line, which he claims has a mean breaking strength of 15 pounds with a standard deviation of 0.5 pounds.
Test the hypothesis that $\mathrm{\mu <!--\; \mu \; -->}$ = 15 pounds against the alternative that $\mathrm{\mu <!--\; \mu \; -->}$ + 15 pounds, if a random sample of 50 lines is tested and found to have a mean breaking strength of 14.8 pounds.
Use a 0.01 level of significance.

Joachim, a third student, noted that the confidence interval in part (a) gives plausible values of the parameter as an interval between two values. He suggested that they develop a one-sided confidence interval because they were only concerned with whether the number 6 was landing face up more often than expected, not less often. The one sided-interval will determine a value L such that all plausible values of p are greater than L. The formula for L is .
(i) Determine the values of z∗ needed to create the one-sided 95 percent confidence interval. Then calculate the value of L.
(ii) Do the results of Joachim’s one-sided confidence interval agree with results of Aurelia’s significance test in part (b)?
Explain your reasoning.

To test for the significance of the phi correlation, one must use the ______________ distribution.
Group of answer choices:
1.t
2. Chi-square
3. z
4. F

In each of Exercises, we have provided a sample mean, sample standard deviation, and sample size. In each case, use the one-mean t-test to perform the required hypothesis test at the 5% significance level.
$x=20,s=4,n=32,HO:\mathrm{\mu <!--\; \mu \; -->}=22,Ha:\mathrm{\mu <!--\; \mu \; -->}<22$

I've analysed newspapers by counting the language distributions of the articles.
The results look like that:
$\overline{)\begin{array}{ccc}\text{Day1}& \text{Day2}& \text{Day3}\\ Economy& Economy& Economy\\ \text{Language 1: }0.35& 0.30& 0.90\\ \text{Language 2: }0.11& 0.10& 0.00\\ \text{Language 3: }0.54& 0.60& 0.10\\ \text{Sport}& \text{Sport}& \text{Sport}\\ \text{Language 1: }0.40& 0.30& 1.00\\ \text{Language 2: }0.20& 0.20& 0.00\\ \text{Language 3: }0.40& 0.50& 0.00\end{array}}$
I've have already posted another question on that topic (Remove statistical outliers), but here comes my second problem. First of all, I want to remove all statistical outliers from data (e.g. day 3), to make it "clean" (see my other question (other post). After that, I want to terminate which changes in my data are just "noise" and witch are significant changes. But I'm not sure how to do it.
I was thinking of the following approach:
I could calculate the standard deviation (like in my other post) and treat every value outside of it as a "significant change". But I think this will cause a mistake if all my values are slightly increasing or decreasing.
Is there any mathematical technique to find the significant changes in my data?
Thanks in advance.

The log return X on a certain stock investment is an N(μ,σ2) random variable.
A financial analyst has claimed that the volatility σ of the log return on this stock is less than 3 units. A random sample of 11 returns on this stock gave an estimated variance of the log-returns as s2=16.
Assess the analyst's claim by using a significance test at level
α=0.05 to test
H0:σ2≤9 against H1:σ2>9.
Find a two-sided 95% confidence interval for σ2.

I have a question as follows:
Assuming the null hypothesis is true for all cases. If we perform multiple pairwise comparisons at the $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$ level, what is the probability of making type I error with AT LEAST ONE of the tests?
The correct answer is greater than 0.05, and I don't know why. Thanks in advance!

To test for the significance of a Pearson correlation, one must use the ______________ distribution.
Group Answer Choices:
1. F
2. Chi-square
3. z
4. t

A teacher wants to know if more than 15% of her students use online resources to complete their homework assignments instead of doing the homework independently. The teacher would like to carry out a test at the 0.05 significance level of
$H_0:p=0.40$
$H_a:p>0.40$
where p = the true proportion of students that use online resources to complete their homework assignments. One day after school, the teacher asked the 40 students studying in her classroom if they used online resources to complete their homework assignments or if they did their homework independently. Are the conditions to perform the significance test met?
No, the students do not represent a random sample and the Large Counts condition is not satisfied.
No, the students do not represent a random sample even though the Large Counts condition is satisfied.
No, the students represent a random sample, but the Large Counts condition is not satisfied.
Yes, the students represent a random sample and satisfy the Large Counts condition.

Geometric interpretation of the multiplication of complex numbers?
I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the x component and the imaginary part is the y component.
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.
However, is there a geometric interpretation for the multiplication of two complex numbers?
I tried out two test ones, 3+i and −2+3i, which multiply to −9+7i. But no geometrical significance seems to be found.
Is there a geometric significance for the multiplication of complex numbers?

If we look at the solutions for the first problem here and the first problem here, we see both problems are one-tailed tests for the upper-tail. However, in the first paper the rejection region is Z > 1.65, but in the second paper the rejection region is Z > 1.96.
How can this be if both problems have a significance level of 0.05 and both are upper-tailed tests? Shouldn't they both have a rejection region of Z > 1.65? Is the second paper incorrect?

Aurelia, another student, suggested they conduct a significance test to investigate the claim.
She tested the hypotheses $H_0:p=16\text{ versus }{H}_a:p>16$ at the significance level of α=0.05.
She obtained a test statistic of 1.83 with a p-value of 0.033.
Do the results of the significance test agree with the results of Clarke’s confidence interval in part (a)?
Explain your reasoning.

A researcher obtains a Pearson correlation of r = 0.43 for a sample of n = 20 participants. For a two-tailed test, which of the following accurately describes the significance of the correlation?​
a.)The correlation is not significant with either $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 or $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.
b.)​The correlation is significant with either $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 or $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.
c.)The correlation is equal to zero.
d.)The correlation is significant with $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 but not with $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.

The following computer output shows an estimated equation. Here; W: Weekly wage.
MALE: Gender, takes the value of 1 if the worker is male and 0 if female.
EDU: Education Level,
EDU1=1 if the worker has no formal education and 0 otherwise,
EDU2=1 if the worker has primary education and 0 otherwise,
EDU3=1 if the workerhas secondary and high school degree and 0 otherwise,
EDU4=1 if the worker hasbachelor, master, and/or Ph.D. degree and 0 otherwise.
EXP: Experience (thenumber of years being employed).
Dependent Variable: W
Method: Least Squares
Date: 01/02/21 Time: 09:15
Sample: 1 935
Included observations: 935
Variable Coefficient Std. Error t-Statistic Prob.
C 508.7969 49.56471 10.26531 0.0000
MALE 486.2831 20.27897 23.97967 0.0000
EDU2 37.93184 35.58875 1.065838 0.2868
EDU3 153.9977 40.51264 3.801225 0.0002
EDU4 251.2214 40.92353 6.138799 0.0000
EXP 10.28997 2.548076 4.038331 0.0001
R-squared 0.461610 Mean dependent var 957.9455
Adjusted R-squared 0.458713 S.D. dependent var 404.3608
S.E. of regression 297.4973 Akaike info criterion 14.23508
Sum squared resid 82220789 Schwarz criterion 14.26615
Log likelihood -6648.902 Hannan-Quinn criter. 14.24693
F-statistic 159.3033 Durbin-Watson stat 0.446774
Prob(F-statistic) 0.000000
a) Write out the estimated wage model below (use 1-digit for decimal):
b) Check the statistical significance of EXP, and EDU2 at 5 % level respectively(hypothesis tests) below: 6
c) Interpret the coefficient of determination below:
d) Interpret the coefficients of MALE, EDU3, and EXP below:

Significance level for a hypothesis test for a linear regression
Consider linear regression model $Y_i=a+b\cdot <!--\; \cdot \; -->x_i+{\mathrm{ϵ<!--\; ϵ\; -->}}_i$, $i=1,2,3,4,5$ where $a,b\in <!--\; \in \; -->\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, ${\mathrm{ϵ<!--\; ϵ\; -->}}_i$ are iid, normally distributed with mean =0 and variance =9. Consider the hypothesis $H_0:b=0$ with the alternative $H_1:b\ne <!--\; \ne \; -->0$, with the critical region $\{|\stackrel{^<!--\; ^\; -->}{b}|>c\}$, where b^ is a maximum likelihood estimator and c is chosen in such a way that the significance level of a test is equal to 0,05. Calculate c.
Could you help me with this exercise? It is taken from the actuary exam organized in my country. I thought that I am able solve this exercise, however my answer c=0,7528 is wrong, the correct answer is c=1,47.
Edited: the exercise seems very easy, but I'm sure that my method of solution is wrong, as I've seen the similar exercise and my method results with the wrong answer as well.
That's why I've decided to start a bounty, however I do not know how (I can see "question eligible for bounty in 59 minutes" only, not "start a bounty")