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Candy! A machine is supposed to fill bags with an average of 19.2 ounces of candy. The manager of the candy factory wants to be sure that the machine does not consistently underfill or overfill the bags. So the manager plans to conduct a significance test at the $\mathrm{Î\pm }=0.10$significance level of

$$\begin{gathered} H_0:\mathrm{μ}=19.2 \\ {H}_a:\mathrm{μ}notequalto19.2 \end{gathered}$$

where $\mathrm{μ}=$the true mean amount of candy (in ounces) that the machine put in all bags filled that day. The manager takes a random sample of 75 bags of candy produced that day and weighs each bag. Check if the conditions for performing the test are met.

Battery life A tablet computer manufacturer claims that its batteries last an average of 10.5 hours when playing videos. The quality-control department randomly selects 20 tablets from each day’s production and tests the fully charged batteries by playing a video repeatedly until the battery dies. The quality control department will discard the batteries from that day’s production run if they find convincing evidence that the mean battery life is less than 10.5 hours. Here are a dot plot and summary statistics of the data from one day:

a. State appropriate hypotheses for the quality-control department to test. Be sure to define your parameter.

b. Check if the conditions for performing the test in part (a) are met.

Paying high prices? A retailer entered into an exclusive agreement with a supplier who guaranteed to provide all products at competitive prices. To be sure the supplier honored the terms of the agreement, the retailer had an audit performed on a random sample of 25 invoices. The percent of purchases on each invoice for which an alternative supplier offered a lower price than the original supplier was recorded.17 For example, a data value

of 38 means that the price would be lower with a different supplier for 38% of the items on the invoice. A histogram and some numerical summaries of the data are shown here. The retailer would like to determine if there is convincing evidence that the mean percent of purchases for which an alternative supplier offered lower prices is greater than 50% in the population of this company’s invoices.

a. State appropriate hypotheses for the retailer’s test. Be sure to define your parameter.

b. Check if the conditions for performing the test in part (a) are met.

Two-sided test Suppose you want to perform a test of $H_0:\mathrm{μ}=64$

versus $H_a:\mathrm{μ}notequalto64$at the $\mathrm{Î\pm }=0.05$significance level. A random sample

of size $n=25$ from the population of interest yields $\stackrel{¯}{x}=62.8$ and $sx=5.36$

. Assume that the conditions for carrying out the test are met.

a. Explain why the sample result gives some evidence for the alternative hypothesis.

b. Calculate the standardized test statistic and P-value.

Stating hypotheses

a. A change is made that should improve student satisfaction with the parking situation at a local high school. Before the change, $37\%$of students approve of the parking that's provided. The null hypothesis $\mathrm{H}0:\mathrm{p}>0.37H_0:p>0.37$is tested against the alternative Ha: $\mathrm{p}=0.37H_a:p=0.37$

b. A researcher suspects that the mean birth weights of babies whose mothers did not see a doctor before delivery is less than 3000 grams. The researcher states the hypotheses as

$\text{H0:}{x}^{-}=3000{\text{grams}}^{-5}{H}_0:\stackrel{¯}{x}=3000\text{grams}$

Ha: ${\mathrm{x}}^{-}<3000$grams $H_a:\stackrel{¯}{x}<3000$grams

explain what's wrong with the stated hypotheses. Then give correct hypotheses.

One-sided test Suppose you want to perform a test of H0: μ=5 versus Ha: μ<5

at the α=0.05 significance level. A random sample of size n=20 from the population of interest yields x¯=4.7 and sx=0.74 . Assume that the conditions for carrying out the test are met.

a. Explain why the sample result gives some evidence for the alternative hypothesis.

b. Calculate the standardized test statistic and P-value.

Attitudes In the study of older students’ attitudes from Exercise 65, the sample mean SSHA score was 125.7 and the sample standard deviation was 29.8.

a. Calculate the standardized test statistic.

b. Find and interpret the P-value.

c. What conclusion would you make?

Candy! In the study of the candy machine from Exercise 66, the sample mean weight for the bags of candy was 19.28 ounces and the sample standard deviation was 0.81 ounce.

a. Calculate the standardized test statistic.

b. Find and interpret the P-value.

c. What conclusion would you make?

Ending insomnia A study was carried out with a random sample of 10 patients who suffer from insomnia to investigate the effectiveness of a drug designed to increase sleep time. The following data show the number of additional hours of sleep per night gained by each subject after taking the drug.18 A negative value indicates that the subject got less sleep after taking the drug.

a. Is there convincing evidence at the $\mathrm{Î\pm }=0.01$significance level that the average sleep increase is positive for insomnia patients when taking this drug?

b. Given your conclusion in part (a), which kind of mistake—a Type I error or a Type II error—could you have made? Explain what this mistake would mean in context.

A school librarian purchases a novel for her library. The publisher claims that the book is written at a fifth-grade reading level, but the librarian suspects that the reading level is lower than that. The librarian selects a random sample of $40$pages and uses a standard readability test to assess the reading level of each page. The mean reading level of these pages is $4.8$with a standard deviation of $0.8$. Do these data give convincing evidence at the $\mathrm{Î\pm }=0.05$ significance level that the average reading level of this novel is less than $5$?