Exercises: Hypothesis testing

In quality control at a crisp factory, we wish to check whether the mean weight of the packets is really $150$ grams, as stated on the packaging. The population standard deviation is known to be $\sigma = 5$ grams. If it is suspected that the packets may weigh either more or less than stated:

1. a) State 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 bound the acceptance region.

We are conducting 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.

Explain in your own words what it means to make a Type I Error in the context of a study attempting to test whether a new study method improves mathematics marks (where $H_0$ is that the method yields no improvement).

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 hypothesis $H_1$: "The mean performance is higher".

Give reasons explaining what making a Type I error and a Type II error would mean 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 sustainability study, it is known that two years ago $30\ \%$ of young people in a town used public transport to get to school. Following an environmental awareness campaign, we want to test whether this percentage has increased. A random sample of $150$ young people is taken, and it is observed that $54$ of them now use public transport.

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

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

Calculate the p-value (the probability of obtaining a result as extreme as, or more extreme than, the observed result under the null hypothesis) for a two-tailed test. In light of the result, would you reject the health center's claim at a standard significance level of $\alpha = 0.05$?

A study was conducted on the monthly leisure expenditure of university students in a city, assuming it follows a normal distribution with $\sigma = 400$ pesos. A $95\ \%$ confidence interval for the mean was obtained as $(1\,502; 1\,698)$.

Without carrying out new standardization calculations, use the relationship between confidence intervals and hypothesis tests to answer: Would the null hypothesis $H_0: \mu = 1\,400$ pesos be rejected against $H_1: \mu \neq 1\,400$ pesos at a significance level of $\alpha = 0.05$? What if the null hypothesis were $H_0: \mu = 1\,600$ pesos? Justify your answer.

A study on youth mental health indicates that the daily mean time that 2nd Baccalaureate students spend on TikTok is $120$ minutes, with a standard deviation of $30$ minutes. A counsellor at a school believes that at their center, due to the pressure of the EBAU 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 counsellor's belief.
2. b) Calculate the p-value of the obtained sample.
3. c) If a significance level of $1\%$ is used, what conclusion should the counsellor draw? What if the significance level were $5\%$?

In a debate on sustainability, it is claimed that $75\%$ of young people in a city separate plastic packaging from the rest of the rubbish. 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 whether the true proportion is lower than claimed.
2. b) Is the initial claim acceptable for a significance level of $10\%$?
3. c) Suppose that the true proportion were $65\%$. What decision would most likely be made with this test if the sample size were only $20$ people? Reason about the importance of sample size on the power of the test.

We want to investigate whether a new study method improves performance in Mathematics. The historical mean grade in a region is $6.4$ with a standard deviation $\sigma = 1.2$. The new method is applied to an experimental group of $36$ students and a mean grade of $6.9$ is obtained.

1. a) Carry out a one-tailed hypothesis test for a significance level $\alpha = 0.05$ (state $H_0$ and $H_1$, calculate the critical value and the rejection region, and conclude).

1. b) If the $95\%$ confidence interval for the mean of this group were $(6.508 ; 7.292)$, could you have answered the previous part without calculating the test statistic? Explain the relationship between confidence intervals and hypothesis testing.