Exercises: Hypothesis testing

A streaming platform claims that the mean time its users spend watching series is $120$ minutes a day. A group of students suspects that, due to exams, this time has decreased. To carry out a hypothesis test on this claim:

1. a) Clearly define the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$).

1. b) State whether it is a one-tailed or two-tailed test.

In the quality control of a potato chip factory, we want to check if the mean weight of the bags is really $150$ ounces, as indicated on the packaging. It is known that the population standard deviation is $\sigma = 5$ ounces. If it is suspected that the bags may weigh both more and less than indicated:

1. a) Set up the hypotheses $H_0$ and $H_1$.

1. b) If we set a significance level of $\alpha = 0,05$, calculate the critical values $z_{\alpha/2}$ that delimit the acceptance zone.

A sociologist claims that $35\%$ of American teenagers prefer physical format to digital for reading books. To test this claim against the suspicion that the proportion is different, a random sample is chosen. Determine the acceptance region for the sample proportion if a confidence level of $95\%$ is used and the sample size is $n = 400$ people.

We are carrying out a two-tailed hypothesis test for the mean of a normal population with known $\sigma$. After calculating the test statistic with the data from a sample, we obtain a value of $z_{exp} = 2,15$.

If the critical values for a significance level of $\alpha = 0,05$ are $-1,96$ and $1,96$, reason whether we should accept or reject the null hypothesis.

A sports shoe brand claims that $60\%$ of marathon runners use its products. A sports journalist suspects that the proportion is lower. To carry out the test, $100$ runners are surveyed and it is found that $52$ of them use this brand.

1. a) Calculate the test statistic (sample proportion) $\hat{p}$.

1. b) Write the expression for the standardised test statistic $Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 \cdot q_0}{n}}}$ for this case.

We want to check if the mean charging time of a smartphone battery has changed after a software update. The mean time before was $90$ minutes with a population standard deviation of $\sigma = 10$ minutes. A sample of $64$ phones is taken and a mean charging time of $93$ minutes is obtained.

For a significance level of $\alpha = 0,01$ (where the critical value is $z_{\alpha/2} = 2,575$), determine the acceptance zone of the two-tailed test for the population mean.

Explain in your own words what it means to commit a Type I Error in the context of a study attempting to prove whether a new study method improves math scores (where $H_0$ is that the method produces no improvements).

Given a normal population with standard deviation $\sigma = 12$, we want to test $H_0: \mu = 50$ against $H_1: \mu > 50$ with a significance level of $\alpha = 0,05$.

Calculate the critical value of the sample mean $\bar{x}_c$ for a sample of size $n = 36$ using the formula $\bar{x}_c = \mu_0 + z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}$, knowing that for $\alpha = 0,05$ the value of $z_{\alpha} = 1,645$.

A well-known technology influencer claims in one of their videos that $40\ \%$ of American teenagers prefer to use tablets rather than laptops for studying. A student association decides to test this claim and carries out a random survey of $250$ high school students, of which $85$ state their preference for the tablet.

Using a significance level of $\alpha = 0,05$, set up a two-tailed hypothesis test to determine if the influencer's claim can be considered statistically acceptable or if, on the contrary, the true proportion is different.

An online shopping courier company claims that the mean delivery time for its parcels in large cities is $48$ hours, with a known standard deviation of $\sigma = 6$ hours. A group of users suspects that the service is slower than advertised. To check this, they record the delivery times of $36$ randomly chosen orders, obtaining a mean of $50,5$ hours.

State the null and alternative hypotheses for this case and carry out a one-tailed (right-tailed) hypothesis test with a confidence level of $99\ \%$. Is there sufficient evidence to claim that the company takes longer than it claims?

A new model of running shoes designed to increase performance is being tested. The manufacturer sets the null hypothesis $H_0$: "The mean performance with the new model is equal to that of the previous model" against the alternative $H_1$: "The mean performance is higher".

Explain with reasoning what it would mean to commit a Type I error and a Type II error in this practical context. Which of the two errors do you think would be more damaging to the company's brand image if discovered later?

In a study on sustainability, it is known that two years ago $30\ \%$ of young people in a town used public transportation to go to school. Following an environmental awareness campaign, we want to check if this percentage has increased. A random sample of $150$ young people is taken and it is observed that $54$ of them now use public transportation.

From these data, and with a significance level of $5\ \%$, can it be concluded that the campaign has been effective and the proportion of users has increased? Set up the corresponding test.

A soft drink manufacturer claims that each can contains, on average, $330$ fl oz of the product. It is known that the filling of the cans follows a normal distribution with a standard deviation of $\sigma = 5$ fl oz. A consumer inspector takes a random sample of $25$ cans and obtains a mean volume of $327,5$ fl oz.

With a significance level of $1\ \%$, carry out a two-tailed test to verify if the mean content of the cans differs significantly from the value advertised by the manufacturer.

A health center claims that the proportion of smokers in the population of young people aged between 16 and 18 is $15\ \%$. To verify this, a study is carried out on a sample of $400$ young people in that age range, resulting in $72$ of them smoking.

Calculate the p-value (probability of obtaining a result as extreme as or more extreme than the one observed under the null hypothesis) for a two-tailed test. In light of the result, would you reject the health center's claim for a standard significance level of $\alpha = 0,05$?

A study has been carried out on the monthly leisure expenditure of college students in a city, assuming it follows a normal distribution with standard deviation $\$\sigma = 20$. A $95\ \%$ confidence interval of $(75,1; 84,9)$ has been obtained for the mean.

Without performing new standardization calculations, use the relationship between confidence intervals and hypothesis testing to answer: Would the null hypothesis $\$H_0: \mu = 70$ be rejected against $\$H_1: \mu \neq 70$ with a significance level of $\alpha = 0,05$? What if the null hypothesis were $\$H_0: \mu = 80$? Justify your answer.

A streaming platform suspects that the mean daily time its users spend watching content has changed compared to last year, which was $120$ minutes. They assume a normal distribution with $\sigma = 30$ minutes. They take a random sample of $100$ users and observe a mean of $126$ minutes.

Determine the acceptance region and the critical region for a two-tailed test with a significance level of $10\ \%$. What decision should the platform make regarding its suspicion?

A well-known eSports organization claims that $40\%$ of teenagers who play "League of Legends" in the US have spent money on in-game cosmetics (skins). However, a consumer association suspects that this figure is different. To test this hypothesis, they carry out a random survey of $600$ players, of which $216$ admit to having made purchases.

1. a) State the null and alternative hypotheses for a two-tailed test.
2. b) Using a significance level of $\alpha = 0,05$, determine the critical region.
3. c) Can the organization's claim be rejected with the data obtained? Justify your answer by calculating the test statistic.

A study on youth mental health indicates that the mean daily time that high school seniors spend on TikTok is $120$ minutes, with a standard deviation of $30$ minutes. A school counselor believes that in their school, due to the pressure of SAT exams, the mean time is significantly lower. They take a random sample of $50$ students and obtain a mean of $112$ minutes.

1. a) Define the appropriate hypothesis test for the counselor's belief.
2. b) Calculate the p-value of the sample obtained.
3. c) If a significance level of $1\%$ is used, what conclusion should the counselor draw? What if the significance level were $5\%$?

A brand of wireless headphones claims that the battery of its new "Pro-Z" model lasts, on average, $24$ hours. It is known that the battery lifespan follows a normal distribution with a standard deviation of $\sigma = 2,5$ hours. A group of technical testers analyzes $100$ headphones and finds a mean lifespan of $23,4$ hours.

1. a) Carry out a one-tailed hypothesis test to check if the battery lasts less than promised with a significance level of $\alpha = 0,05$.
2. b) Explain, in the context of the problem, what committing a Type II error would consist of.
3. c) What should the minimum sample size be so that, with a mean of $23,4$, the result is significant at $1\%$?

In a debate on sustainability, it is claimed that $75\%$ of young people in a city separate plastic packaging from the rest of the trash. To verify this, a sample of $150$ young people is selected and it is observed that $105$ of them recycle correctly.

1. a) Set up the hypothesis test to check if the true proportion is lower than claimed.
2. b) Is the initial claim acceptable for a significance level of $10\%$?
3. c) Suppose the true proportion were $65\%$. What decision would most likely be made with this test if the sample were only $20$ people? Reason about the importance of sample size on the power of the test.

A manufacturer of electronic components for smartphones guarantees that the percentage of defective parts in its production is, at most, $2\%$. An external quality control decides to sample $2500$ components to verify this guarantee. After the analysis, $62$ defective parts are found.

1. a) Formulate the hypotheses for a right-tailed test.
2. b) Calculate the value of the test statistic.
3. c) Determine the critical significance level (p-value). In light of the result, do you think the manufacturer should review its production process? Justify your answer.

We want to investigate whether a new study method improves performance in math. The historical mean score in a region is $6,4$ with a standard deviation of $\sigma = 1,2$. The new method is applied to an experimental group of $36$ students and a mean score of $6,9$ is obtained.

1. a) Carry out a one-tailed hypothesis test for a significance level of $\alpha = 0,05$.
2. b) If the $95\%$ confidence interval for the mean of this group were $(6,508 ; 7,292)$, could you have answered the previous section without calculating the test statistic? Explain the relationship between confidence intervals and hypothesis testing.

A trade school claims that the mean starting salary of its graduates is $\$1550$ per month. It is known that salaries follow a normal distribution with standard deviation $\$\sigma = 210$. A trade union selects a sample of $49$ graduates and obtains a mean salary of $\$1480$.

1. a) Test the school's hypothesis against the union's suspicion that the salary is lower, using $\alpha = 0,01$.
2. b) Calculate the probability of committing a Type I error if we decide to reject the null hypothesis for any sample mean lower than $\$1500$.
3. c) How would an increase in the population standard deviation affect the acceptance region?

A streaming platform considers a user to be "loyal" if they watch more than $15$ hours of content per month. The company maintains that $60\%$ of its current subscribers meet this criterion. For a market study, the profiles of $400$ randomly chosen users are analyzed and it is found that $224$ are loyal users.

1. a) Set up a two-tailed test for the proportion of loyalty with $\alpha = 0,05$.
2. b) What is the maximum allowable error of the proportion so that the null hypothesis is not rejected with the sample obtained?
3. c) If we change the significance level to $\alpha = 0,01$, would the decision change? Reason about how the strictness of the test affects the possibility of rejecting a claim.