Pregunta
upstudy study bank question image url

In a random sample of 15 -year-olds, the mean height was 65 inches. The sampling distribution was normal with a standard error of 0.50 inches. A. [ 3 points] Construct a \( 90 \% \) confidence interval to estimate the population mean. Round your answer to 3 decimal places. Use the general form of a confidence interval: sample statistic \( \pm z^{*}(S E) \) Show your work.

Ask by Simmons Mcdonald. in the United States
Mar 11,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The 90% confidence interval for the population mean height is [64.178, 65.823].

Solución

**Step 1: Identify the given information** - Sample mean: \( \bar{x} = 65 \) inches - Standard error: \( SE = 0.50 \) inches - Confidence level: \( 90\% \) **Step 2: Find the critical value \( z^{*} \) for a \( 90\% \) confidence level** For a \( 90\% \) confidence interval, the tail probability is \[ \alpha = 1 - 0.90 = 0.10 \] Since the confidence interval is symmetric, each tail has a probability of \[ \frac{\alpha}{2} = 0.05. \] The critical value \( z^{*} \) corresponds to the \( 95^{th} \) percentile of the standard normal distribution, which is \[ z^{*} \approx 1.645. \] **Step 3: Compute the margin of error** The margin of error is calculated as: \[ ME = z^{*} \times SE = 1.645 \times 0.50 = 0.8225. \] **Step 4: Construct the confidence interval** The general form of the confidence interval is: \[ \bar{x} \pm z^{*}(SE) \] Substitute the known values: \[ 65 \pm 0.8225 \] Thus, the lower bound is: \[ 65 - 0.8225 = 64.1775, \] and the upper bound is: \[ 65 + 0.8225 = 65.8225. \] **Step 5: Round the answer to 3 decimal places** Lower bound: \[ 64.1775 \approx 64.178 \] Upper bound: \[ 65.8225 \approx 65.823. \] The \( 90\% \) confidence interval to estimate the population mean is: \[ [64.178, \ 65.823]. \]

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Beyond the Answer

To construct a \( 90\% \) confidence interval for the population mean height of 15-year-olds, we start with the sample statistics. The sample mean (\( \bar{x} \)) is \( 65 \) inches and the standard error (SE) is \( 0.50 \) inches. First, we need to find the critical value (\( z^{*} \)) for a \( 90\% \) confidence level. For a \( 90\% \) confidence interval, the area in each tail of the standard normal distribution is \( 0.05 \). Using the standard normal (Z) table or calculator, we find that \( z^{*} \approx 1.645 \). Now we can calculate the confidence interval using the formula: \[ \text{Confidence Interval} = \bar{x} \pm z^{*} \times SE \] Substituting in the values: \[ \text{Confidence Interval} = 65 \pm 1.645 \times 0.50 \] Calculating the margin of error: \[ 1.645 \times 0.50 = 0.8225 \] Now we can construct the interval: \[ \text{Lower Limit} = 65 - 0.8225 = 64.1775 \] \[ \text{Upper Limit} = 65 + 0.8225 = 65.8225 \] Finally, rounding to three decimal places gives us: \[ \text{Confidence Interval} = (64.178, 65.823) \] Thus, the \( 90\% \) confidence interval for the population mean height of 15-year-olds is \( (64.178, 65.823) \).

preguntas relacionadas

Latest Statistics Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad