A nutritionist wants to compare the effects of two different diets on weight loss. After $8$ weeks, the following statistics are recorded:

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$. At the $0.05$ significance level, test the claim that there is no significant difference in mean weight loss between the two diets.

An ecologist records the average daily temperature in degrees Celsius and the number of bird species observed in $7$ different locations: $(12,5),(15,7),(18,9),(10,4),(20,11),(14,6),(16,8)$. At a significance level of $\alpha$ equals $0.05$, is there evidence of a linear correlation between temperature and the number of bird species? Use the critical value approach.

A tire manufacturer claims that the mean mileage of its competitor’s tires is less than $40,000$ miles. You are asked to conduct a hypothesis test to evaluate this claim. How would you state the null and alternative hypotheses if you were representing the competitor and wanted to reject the claim?

A nutritionist claims that the mean fat content in a brand of yogurt is less than $4$ grams per serving. If the null hypothesis is not rejected, what does this imply?

Generate $100$ independent simple random samples of size $n=25$ from a normal population with mean $\mu =0$ and known standard deviation $\sigma =5$. Perform two-sided tests of $H_0:\mu =0$ versus $H_1:\mu \ne 0$ at a significance level $\alpha =0.01$. What are the approximate expected number of rejections and the standard deviation of the number of rejections under the null hypothesis?

A smartphone manufacturer claims that the average battery life of its new model is at least $20.0$ hours. A consumer group tests a random sample of $16$ phones and finds a mean battery life of $18.7$ hours with a standard deviation of $2.4$ hours. At $\alpha =0.05$, is there sufficient evidence to reject the manufacturer's claim? Assume the population is normally distributed.

A health study asks participants, “Do you exercise in the morning or evening?” Responses are 'morning', 'evening', and 'never'. Use a sign test to test if the proportion exercising in the morning equals the proportion exercising in the evening. Assign $+$ to 'morning', $-$ to 'evening', and $0$ to 'never'. Suppose there are $12$ 'morning', $17$ 'evening', and $6$ 'never' responses. Use $\alpha =0.05$. What is the $p$-value?

A vaccine is tested on $950$ volunteers, and $38$ of them report mild fever as a side effect. The manufacturer claims that at least $7$ percent of vaccinated individuals experience mild fever. Use a $0.05$ significance level to test the claim. Answer the following:

a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?

A researcher claims that at least $55\%$ of adults prefer online shopping. In a sample of $120$ adults, $60$ say they prefer online shopping. Can a normal sampling distribution be used? If so, test the claim at $\alpha =0.05$.

A manufacturer produces bottles of olive oil labelled as containing $500\text{ mL}$. A quality control specialist wants to ensure the consistency of the filling process. From a simple random sample of $30$ bottles, the following statistics are obtained: $n=30$, $\bar{x}=498.7\text{ mL}$, $s=2.05\text{ mL}$.

Assume the population of fill volumes is normally distributed. At the $0.05$ significance level, test the claim that the standard deviation of the fill volume is less than $3.00\text{ mL}$.

The monthly rent (in dollars) paid by $12$ randomly selected residents in a city is listed below:

$950$, $1020$, $990$, $980$, $1005$, $1015$, $995$, $1030$, $1010$, $1000$, $985$, $970$

At the $\alpha =0.05$ level of significance, is there sufficient evidence to conclude that the standard deviation of monthly rent is different from $\text{\$}25$? Assume the population is normally distributed.

Find the critical value and rejection region for a left-tailed $z$-test where $\alpha =0.025$.

Determine the critical value and rejection region for a left-tailed $t$-test with $\alpha =0.025$ and $n=41$.

A quality control analyst wants to determine whether the variability in the weight of small snack packets has decreased from what was previously known. She takes a random sample of $7$ packets and performs a left-tailed chi-square test at the $1\%$ significance level. What is the critical value and rejection region for this left-tailed chi-square test?