Eroxl's Notes
Web Work 8 (STAT 251)

Problem 1

In a study to estimate the proportion of residents in a city that support the construction of a new bypass road in the vicinity, a random sample of 1225 residents were polled. Let denote the number in the sample who supported the proposal. To estimate the true proportion in support of the plan, we can compute The estimator has what bias.

  • [ ] A.
  • [x] B.
  • [ ] C. , so unbiased
  • [ ] D.
  • [ ] E.

Problem 2

The time in hours for a worker to repair an electrical instrument is a Normally distributed random variable with a mean of and a standard deviation of 50. The repair times for 12 such instruments chosen at random are as follows:

(a). Find a 95% Confidence Interval for . For both Sides of the Bound, Leave Your Answer with 3 Decimal place

(b). Find the Least Number of Repair times Needed to Be Sampled in order to Reduce the Width of the Confidence Interval to below 31 Hours

Problem 3

Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of 31 vehicles was 82.72 km/h, with the sample standard deviation being 4.61 km/h.

We will assume that the speeds are Normally distributed, and the police are interested in the mean speed.

(a). Since the variance of the Underlying Normal Distribution is not Known, Inference here Would Involve the Distribution. How Many Degrees of Freedom Would the Relevant Distribution Have?

(b). Create a 95% Confidence Interval for the Mean Speed of Vehicles Crossing the Bridge. Give the Upper and Lower Bounds to Your Interval, Each to 2 Decimal Places

(c). The Police Hypothesized that the Mean Speed of Vehicles over the Bridge Would Be the Speed Limit, 80 km/h. Taking a Significance Level of 5%, what Should Infer about This Hypothesis?

Since our statistic is greater than we reject the hypothesis.

  • [ ] A. We should reject the hypothesis since the sample mean was not 80 km/h.
  • [ ] B. We should not reject the hypothesis since the sample mean is in the interval found in (b).
  • [x] C. We should reject the hypothesis since 80 km/h is not in the interval found in (b).
  • [ ] D. We should reject the hypothesis since 80 km/h is in the interval found in (b).
  • [ ] E. We should not reject the hypothesis since 80 km/h is in the interval found in (b).

(d). Decreasing the Significance Level of the Hypothesis Test above Would

  • [ ] A. either increase or decrease the Type I error probability.
  • [ ] B. not change the Type I error probability.
  • [ ] C. increase the Type I error probability.
  • [x] D. decrease the Type I error probability.
  • [ ] E. not change the Type II error probability.

Problem 4

Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was 25.84 (in 10 miles) and the variance was 3.76 (in 10 miles). For the cars with brand B, the corresponding statistics were 24.64 (in 10 miles) and the variance was 8.80 (in 10 miles) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands.

(a). Calculate the Pooled variance to 3 Decimal Places. During Intermediate Steps to Arrive at the Answer, Make Sure You Keep as Many Decimal Places as Possible so that You Can Achieve the Precision Required in This Question