STAT 2332 Exam 4

A sociologist studying marriage in Spain and Italy wanted to compare how old, on average, women in each country are when they first get married. The sociologist obtained a random sample of married women from each country. Here is a summary of the ages at first marriage for the women in each sample: Spain Italy x - bar 29.5 28.8 z 2.5 3.6 N 84 73 The researcher wants to use these results to test H0: μS − μI = 0 versus Ha: μS − μI > 0.What is the P-value associated with these sample results?

0.05 < p-val < 0.10

Which of the following chi square values would likely result in failing to reject the null hypothesis?

0.35

What is the expected value for all cells, if we assume that this is a fair six-sided die? total "face sided value" = 6 total observed = 96

16

One way to measure a person's fitness is to measure their body fat percentage. Average body fat percentages vary by age, but according to some guidelines, the normal range for men is 15-20% body fat, and the normal range for women is 20-25% body fat. The sample data is from a group of men and women who did workouts at a gym three times a week for a year. Then, their trainer measured the body fat. The table below shows the data. Group Body Fat Percentages Men 13.3 6.0 20.0 8.0 14.0 19.0 18.0 25.0 16.0 24.0 15.0 1.0 15.0 Women 22.0 16.0 21.7 21.0 30.0 26.0 12.0 23.2 28.0 23.0 Perform a test of significance to determine whether the underlying populations of men and women at the gym have the same mean body fat. Assume the populations of men and women at this gym have equal variances and use 1% significance level.

2-sample t- test, pooled variances, t=-2.80, p-val = 0.0107. Fail to reject the null hypothesis. Men and women at the gym have the same mean body fat.

(Chi-square distributions - GOF) The table shows the number or babies born on each day of the week. How many degrees of freedom will be present? 7 rows & 2 columns

6

What is the χ2 calculated value for the following situation. Coke Pepsi Total Male 19 6 25 Female. 10. 15. 25 Total. 29. 21. 50

6.65

The contingency table below shows the results of a random sample of 300 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliation. Assuming the row and column classification are independent, find an estimate for the expected number of people who are Democrats and Approve the bill (E21). Party Approve Disapprove No Opinion Republican 63 30. 21 Democrat. 75 36 27 Independent 15 24 9

70.4

A chi-square test is used to test whether a 0 to 9 spinner is "fair" (that is, the outcomes are all equally likely). The spinner is spun 100 times, and the results are recorded. The degrees of freedom for the test will be

9

Which of the following is true about the t-distribution?

All of the above (n<30; normal distribution; something else)

(Extra credit) The diameter of cork of a Wine bottle is supposed to be 2 cm. If the cork is either too large or too small, it will not fit in the bottle. The manufacturer measures the diameter in a random sample of 50 bottles and construct a 95% confidence interval: (1.75cm, 1.95cm). Is there evidence that the true mean diameter has moved away from the target?Which of the following is TRUE?

At 5% significance level, there is strong evidence that the true mean diameter has moved away from the target.

Is the type of television show that a person chooses to watch related to the brand of deodorant that a person uses? 1500 people were surveyed.

Chi-Square Test for Independence/Association

In a study of the television viewing habits of children, a developmental psychologist selects a random sample of 300 first graders - 100 boys and 200 girls. Each child is asked which of the following TV programs they like best: The Lone Ranger, Sesame Street, or The Simpsons. Viewing Preferences Lone Ranger. Sesame Street. The Simpsons. Total Boys. 50 30. 20 100 Girls 50. 80. 70. 200 Total. 100. 110. 90 300 Do the boys' preferences for these TV programs differ significantly from the girls' preferences? Which test should we use?

Chi-square, Homogeneity

(𝜒2 distributions - which test) A group of 276 healthy men and women were grouped according to their number of relationships. They were then exposed to a virus that caused colds. The data is summarized in the table below. Does the data provide sufficient evidence to indicate that susceptibility to colds is affected by the number of relationships you have? 3 or less 4-5 6 or more Cold 49 43 34 No cold 31 47 62 Which test should we use?

Chi-square, Independence

The observed frequencies of genotypes A, B and C of 100 progeny from a genetic cross are Oi: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.Under the model, the expected frequencies (Ei) are:

Ei: 25, 50, 25

Continuation. What is an appropriate conclusion at the α=0.10 significance level? Fill in the blanks and choose an answer

Fail to reject H0. This isn't enough evidence to conclude that the mean weight is different than 0.5 lbs

A sample survey obtains information on characteristics of book readers. The following data were obtained from random samples of book-readers, non-book readers and non-readers 16 years old and over in an effort to determine whether or not the proportions of book readers, non-book readers, and non-readers are the same for each income bracket. State the hypotheses for a chi-squared test of homogeneity between these 3 populations:

H0: The classifications have the same distribution for each household income bracket.Ha: The classifications do not have the same distribution for each household income bracket.

The mean fineness µ, of yarn is expected to be greater than the standard value of 5 units. To test this claim, the factory graded 16 specimens of the yarn and found the sample average, x-bar, to be 5.9 units.What are the null and alternative hypotheses being tested?

H0: µ = 5; H1: µ > 5

In their advertisements, a new diet program would like to claim that their methods result in a mean weight loss of more than ten pounds in two weeks. In order to determine if this is a valid claim, they hire an independent testing agency that then selects twenty-five people to be placed on this diet. The agency should be testing the null hypothesis H0: = 10 and the alternative hypothesis.

Ha: μ > 10

(Continuation) Assume the coin is actually unfair. Which of the following is correct under 0.10 significant level?

If P-value is 0.11 > 0.10, then this is Type II error

According to the daily planet 63% of students commute to school in the US. Researchers at UTD if this percentage stays true for the student body at UTD. The researchers take a random sample of 2500 students and find that 1620 students commute to school. Does the percentage of students who commute to school at UTD differ from the percentage claimed by the daily planet with a significance level of 0.05.

No, there wasn't a significant difference between UTD's percentage and the daily planets.

In the calculation of the chi-squared test statistic, what does the "O" represent?

Observed Frequency

A pharmaceutical laboratory was concerned that 3 of its drugs developed to reduce LDC (bad cholesterol) were different with respect to the side effect - abdominal pain. So they collected the following data on the 3 groups and asked if the patients had (or not) abdominal pain: Group1 Group 2 Group 3 Row Total #people w/pain 151 16 240 407 #people w/o pain 1,532 152 1,847 3,531 Column Total. 1,683 168 2,087 3,938 We want to know if the proportion of subjects in each group who experience abdominal pain is different at 1% significance level.H0: p1 = p2 = p3HA: The population's proportions are not all equal

P-value > alpha. This result is not statistically significant, and we don't have sufficient evidence to reject H0. The population's proportions are all equal.

Apgar score is a score between 0 and 10 that gives a measure of the physical condition of a newborn infant. Researchers collected the Apgar scores of 20 pairs of identical twins.The researchers wanted to test if their results suggest a significant difference in the Apgar score between the first-born twin and the second-born twin. Assume that the necessary conditions for inference were met.Which of these is the most appropriate test and alternative hypothesis?

Paired t-test with Ha: μdifference≠0

Suppose the P-value for a hypothesis test is 0.0304. Using a = 0.05, what is the appropriate conclusion?

Reject the null

Suppose a consumer product researcher wanted to find out whether a Sharpie lasted longer than the manufacturer's claim that their Sharpies could write continuously for a mean of 14 hours. The researcher tested 25 Sharpies and recorded the number of continuous hours each Sharpie wrote before drying up. Test the hypothesis that Sharpies can write for more than a mean of 14 continuous hours. Following are the summary statistics: x-bar=14.5 hours, s=1.0 hours At the 5% significance level, t = 2.5; p = 0.0098. State your conclusion about the original claim.

Reject the null hypothesis; there is strong evidence to suggest that the Sharpies last longer than a mean of 14 hours

Papa John's reports that the average time to cook at pizza is 4 minutes. A few avid pizza lovers conduct a study of 19 pizzas being cooked. The study finds that the mean time to cook a pizza is 3.34 minutes and the standard deviation is 1.32 minutes. What are the test statistic (z or t) and the p-value if the pizza lovers want to know if it actually takes less time, on average, to cook a pizza?

T= -2.18, p-val = 0.0214

If a research group performs a hypothesis test and gets a p-value of 0.0345. Given that they used a significance level 0.05, which of the following statements is correct?

The p-value is < α, so they reject the null hypothesis and have a risk of Type 1 error.

You plan to perform a hypothesis test with a level of significance of 0.05. What's the effect on the probability of committing a Type I error if the sample size is increased?

The probability of committing a Type I error is unchanged

Ava's favorite candy is available with or without peanuts. She wonders if the color distribution is the same for both varieties, so she buys some bags of each variety (she's willing to treat these as separate random samples). Here is a summary of the candies in each sample and the results from a chi-squared test:Chi-square test: Color vs. variety Peanuts. No peanuts Red. 39. 33 Expected. 37. 34.91 Yellow. 377 40 Expected. 39.67 37.33 Green 26 23 Expected 25.24 23.76 χ2=0.619, DF=2, P-value=0.734. Assume that all conditions for inference were met.At the α=0.05 significance level, what is the most appropriate conclusion to draw from this test?Hint: decide whether this is a test for independence or homogeneity first.

This isn't enough evidence to say that the distribution of color differs between these varieties.

A study had subjects read a passage and take a comprehension test on what they read. The researchers randomly assigned subjects so that half of them listened to music with lyrics while reading, while the other half listened to music without lyrics while reading. Here are their scores on the comprehension test: Music with lyrics 75 62 65 41 54 Music without lyrics 64 71 88 77 62 The researchers want to test if these results represent a significant difference in the average score between the groups. Assume that the necessary conditions for inference were met.Which of these is the most appropriate test and alternative hypothesis?

Two-sample t-test with Ha: μlyrics ≠ μwithout

If we reject H0 when in fact H0 is true, this is a _________ error.

Type I

If we fail to reject H0 when in fact Ha is true, this is a _________ error.

Type II

The average diastolic blood pressure of a group of women in Texas suffering from high blood pressure is 96 mmHg. During a clinical trial, the women received a medication, the is supposed to lower their blood pressure. After two months, the researcher wants to perform a hypothesis test to determine whether the medication is effective. He obtained a P-value is 0.3. But the truth is that the average diastolic blood pressure of the women in Texas has decreased. Assume alpha=0.05. What type of error he has done?

Type II error. Failing to reject the hypothesis that μ = 96 mmHg when in fact μ < 96 mmHg.

One-hundred people are selected at random and tested for colorblindness to determine whether gender and colorblindness are independent. The following counts were obtained: Colorblind Not Colorblind Total Male 7 53 60 Female 1 39 40 Overall 8 92 100 If possible, perform a Test of Hypothesis at the alpha = 0.05 significance level:

We cannot perform the test as the requirement Eij ≥ 5 is not met.

An Air Force vase mess hall has received a shipment of 10,000-gallon size cams of cherries. The supplier claims that the average amount of liquid is 0.25 gallons per can. A government inspector took a random sample of 100 cans and found that the average liquid content to be 0.28 gallons per can with a standard deviation of 0.10. Does this indicate that the supplier's claim is too low? (Use a 5% level of significance)

Yes, there was a significant difference found

5. A hypothesis test tests the null hypothesis Ho: μ = 20 against the alternative hypothesis Ha: μ ≠ 20 at a significance level of alpha. If the same procedure is used to test Ho: μ = 20 against Ha: μ > 20 with the same data then the p-value will:

decrease

What must be true about the expected values in a chi square test?

greater than or equal to 5

(𝜒2 distributions - which test) A group of 276 healthy men and women were grouped according to their number of relationships. They were then exposed to a virus that caused colds. The data is summarized in the table below. Does the data provide sufficient evidence to indicate that susceptibility to colds is affected by the number of relationships you have? 3 or less 4-5 6 or more Cold 49 43 34 No cold 31 47 62 What is the P-value associated with these sample results?

p-val < 0.01

The shape of the Chi-Square distribution is _________.

right skewed

A coin is tossed 10000 times, and it lands heads 5080 times. We want to determine the chance of heads equal to 50% (H0 : The coin is fair, H0 : The coin is not fair). Which of the following is correct? (Hint: perform the test.)

under the 0.10 significant level, we fail to reject H0

The table shows the number or babies born on each day of the week. Is there evidence that births are more likely on specific days of the week? You need to perform a Chi-square test of Goodness of Fit.

yes, the p value is less than .05

A fast-food company advertises that the pre-cooked weight for its half-pound burgers is, on average, 0.5 lbs. Alvaro is in charge of a quality control test of H0: μ = 0.5 lbs versus Ha: μ ≠ 0.5 lbs, where μ is the mean weight of all burgers in a batch. Alvaro took a random sample of 30 burgers from a batch and found a mean weight of 0.49 lbs and a sample standard deviation of 0.04 lbs. Based on these results, he calculated a test statistic of ____(z or t) = _____(value here) and a P-value of approximately ________.

z = -1.37, p-val = 17.09%

Suppose a consumer product researcher wanted to find out whether a highlighter lasted longer than the manufacturer's claim that their highlighters could write continuously for 14 hours. The researcher tested 40 highlighters and recorded the number of continuous hours each highlighter wrote before drying up. Test the hypothesis that the highlighters wrote for more than 14 continuous hours. Following are thesummary statistics: =14.5 hours, s =1.2 hoursReport the test statistic, p-value, your decision regarding the null hypothesis. At the 5% significance level, state your conclusion about the original claim. Round all values to the nearest thousandth.

z = 2.635; p = 0.004; Reject the null hypothesis; there is strong evidence to suggest that the highlighters last longer than 14 hours.